# Basic Concepts in the Geometry of Banach Spaces

Basic Concepts in the Geometry of Banach Spaces

Basic Concepts in the Geometry of Banach Spaces

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Preface<br />

1. Introduction<br />

The aim <strong>of</strong> this Handbook is to present an overview <strong>of</strong> <strong>the</strong> ma<strong>in</strong> research directions and results<br />

<strong>in</strong> <strong>Banach</strong> space <strong>the</strong>ory obta<strong>in</strong>ed dur<strong>in</strong>g <strong>the</strong> last half century. The scope <strong>of</strong> <strong>the</strong> <strong>the</strong>ory,<br />

hav<strong>in</strong>g widened considerably over <strong>the</strong> years, now has deep and close ties with many areas<br />

<strong>of</strong> ma<strong>the</strong>matics, <strong>in</strong>clud<strong>in</strong>g harmonic analysis, complex analysis, partial differential equations,<br />

classical convexity, probability <strong>the</strong>ory, comb<strong>in</strong>atorics, logic, approximation <strong>the</strong>ory,<br />

geometric measure <strong>the</strong>ory, operator <strong>the</strong>ory, and o<strong>the</strong>rs. In choos<strong>in</strong>g a topic for an article <strong>in</strong><br />

<strong>the</strong> Handbook we considered both <strong>the</strong> <strong>in</strong>terest <strong>the</strong> topic would have for non-specialists as<br />

well as <strong>the</strong> importance <strong>of</strong> <strong>the</strong> topic for <strong>the</strong> core <strong>of</strong> <strong>Banach</strong> space <strong>the</strong>ory, which is <strong>the</strong> study<br />

<strong>of</strong> <strong>the</strong> geometry <strong>of</strong> <strong>in</strong>f<strong>in</strong>ite dimensional <strong>Banach</strong> spaces and n-dimensional normed spaces<br />

with n f<strong>in</strong>ite but large (local <strong>the</strong>ory).<br />

Many <strong>of</strong> <strong>the</strong> lead<strong>in</strong>g experts on <strong>the</strong> various aspects <strong>of</strong> <strong>Banach</strong> space <strong>the</strong>ory have written<br />

an exposition <strong>of</strong> <strong>the</strong> ma<strong>in</strong> results, problems, and methods <strong>in</strong> areas <strong>of</strong> <strong>the</strong>ir expertise. The<br />

enthusiastic response we received from <strong>the</strong> community was gratify<strong>in</strong>g, and we are deeply<br />

appreciative <strong>of</strong> <strong>the</strong> considerable time and effort our contributors devoted to <strong>the</strong> preparation<br />

<strong>of</strong> <strong>the</strong>ir articles.<br />

Our expectation is that this Handbook will be very useful as a source <strong>of</strong> <strong>in</strong>formation and<br />

<strong>in</strong>spiration to graduate students and young research workers who are enter<strong>in</strong>g <strong>the</strong> subject.<br />

The material <strong>in</strong>cluded will be <strong>of</strong> special <strong>in</strong>terest to researchers <strong>in</strong> <strong>Banach</strong> space <strong>the</strong>ory who<br />

may not be aware <strong>of</strong> many <strong>of</strong> <strong>the</strong> beautiful and far reach<strong>in</strong>g facets <strong>of</strong> <strong>the</strong> <strong>the</strong>ory. We ourselves<br />

were surprised by <strong>the</strong> new light thrown by <strong>the</strong> Handbook on directions with which<br />

we were already basically familiar. We hope that <strong>the</strong> Handbook is also valuable for ma<strong>the</strong>maticians<br />

<strong>in</strong> related fields who are <strong>in</strong>terested <strong>in</strong> learn<strong>in</strong>g <strong>the</strong> new directions, problems,<br />

and methods <strong>in</strong> <strong>Banach</strong> space <strong>the</strong>ory for <strong>the</strong> purpose <strong>of</strong> transferr<strong>in</strong>g ideas between <strong>Banach</strong><br />

space <strong>the</strong>ory and o<strong>the</strong>r areas.<br />

Our <strong>in</strong>troductory article, "<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces", is <strong>in</strong>tended<br />

to make <strong>the</strong> Handbook accessible to a wide audience <strong>of</strong> researchers and students. In this<br />

chapter those concepts and results which appear <strong>in</strong> most aspects <strong>of</strong> <strong>the</strong> <strong>the</strong>ory and which go<br />

beyond material covered <strong>in</strong> most textbooks on functional and real analysis are presented<br />

and expla<strong>in</strong>ed. Some <strong>of</strong> <strong>the</strong> results are given with an outl<strong>in</strong>e <strong>of</strong> pro<strong>of</strong>; virtually all are<br />

proved <strong>in</strong> <strong>the</strong> books on <strong>Banach</strong> space <strong>the</strong>ory referenced <strong>in</strong> <strong>the</strong> article. In pr<strong>in</strong>ciple, <strong>the</strong><br />

basic concepts article conta<strong>in</strong>s all <strong>the</strong> background needed for read<strong>in</strong>g any o<strong>the</strong>r chapter <strong>in</strong><br />

<strong>the</strong> Handbook.<br />

Each article past <strong>the</strong> basic concepts one is devoted to one specific direction <strong>of</strong> <strong>Banach</strong><br />

space <strong>the</strong>ory or its applications. Each article conta<strong>in</strong>s a motivated <strong>in</strong>troduction as well as

vi<br />

Preface<br />

an exposition <strong>of</strong> <strong>the</strong> ma<strong>in</strong> results, methods, and open problems <strong>in</strong> its specific direction.<br />

Most have an extensive bibliography. Many articles, even <strong>the</strong> basic concepts one, conta<strong>in</strong><br />

new pro<strong>of</strong>s <strong>of</strong> known results as well as expositions <strong>of</strong> pro<strong>of</strong>s which are hard to locate <strong>in</strong><br />

<strong>the</strong> literature or are only outl<strong>in</strong>ed <strong>in</strong> <strong>the</strong> orig<strong>in</strong>al research papers.<br />

The format <strong>of</strong> <strong>the</strong> chapters is as varied as <strong>the</strong> personal scientific styles and tastes <strong>of</strong> <strong>the</strong><br />

contributors. In our view this makes <strong>the</strong> Handbook more lively and attractive.<br />

There are many strong and <strong>in</strong>tricate <strong>in</strong>terconnections between <strong>the</strong> different subjects<br />

treated <strong>in</strong> <strong>the</strong> various articles, so some articles conta<strong>in</strong> many cross references to o<strong>the</strong>r<br />

articles. But, as mentioned above, no article depends on anyth<strong>in</strong>g past <strong>the</strong> material treated<br />

<strong>in</strong> <strong>the</strong> basic concepts article. Because <strong>of</strong> this lack <strong>of</strong> <strong>in</strong>terdependence, we chose to order <strong>the</strong><br />

chapters after <strong>the</strong> basic concepts article alphabetically accord<strong>in</strong>g to <strong>the</strong> first author <strong>of</strong> <strong>the</strong><br />

chapter. We <strong>in</strong>cluded <strong>in</strong> volume 1 <strong>the</strong> chapters which came first <strong>in</strong> this order among those<br />

articles which were ready by <strong>the</strong> deadl<strong>in</strong>e we set for this volume. The second volume <strong>of</strong><br />

this Handbook will conta<strong>in</strong> <strong>the</strong> chapters which come later <strong>in</strong> <strong>the</strong> alphabetical order as well<br />

as those chapters which were f<strong>in</strong>ished only after <strong>the</strong> deadl<strong>in</strong>e for volume 1.<br />

This volume ends with a subject <strong>in</strong>dex and an author <strong>in</strong>dex for <strong>the</strong> material <strong>of</strong> <strong>the</strong> present<br />

volume. Volume 2 will conta<strong>in</strong> a subject and author <strong>in</strong>dex for <strong>the</strong> entire Handbook.<br />

William B. Johnson and Joram L<strong>in</strong>denstrauss

List <strong>of</strong> Contributors<br />

Abramovich, Yu.A., Indiana University- Purdue University, Indianapolis, IN (Ch. 2)<br />

Aliprantis, C.D., Purdue University, West Lafayette, IN (Ch. 2)<br />

Alspach, D., Oklahoma State University, Stillwater, OK (Ch. 3)<br />

Ball, K., University College London, London (Ch. 4)<br />

Bourga<strong>in</strong>, J., Institute for Advanced Study, Pr<strong>in</strong>ceton, NJ (Ch. 5)<br />

Burkholder, D., University <strong>of</strong> lll<strong>in</strong>ois, Urbana, IL (Ch. 6)<br />

Casazza, E, University <strong>of</strong> Missouri, Columbia, MO (Ch. 7)<br />

Davidson, K.R., University <strong>of</strong> Waterloo, Waterloo (Ch. 8)<br />

Delbaen, E, E.T.H., Zurich (Ch. 9)<br />

Deville, R., University <strong>of</strong> Bordeaux, Bordeaux (Ch. 10)<br />

Diestel, J., Kent State University, Kent, OH (Ch. 11)<br />

Dilworth, S.J., University <strong>of</strong> South Carol<strong>in</strong>a, Columbia, SC (Ch. 12)<br />

Enflo, E, Kent State University, Kent, OH (Ch. 13)<br />

Figiel, T., Institute <strong>of</strong> Ma<strong>the</strong>matics, Sopot (Ch. 14)<br />

Fonf, V.E, Ben Gurion University, Beer-Sheva (Ch. 15)<br />

Gamel<strong>in</strong>, T., The University <strong>of</strong> California, Los Angeles, CA (Ch. 16)<br />

Ghoussoub, N., University <strong>of</strong> British Columbia, Vancouver (Ch. 10)<br />

Giannopoulos, A.A., University <strong>of</strong> Crete, Heraklion (Ch. 17)<br />

Godefroy, G., Universite Paris VI, Paris (Ch. 18)<br />

Jarchow, H., University <strong>of</strong> Zurich, Zurich (Ch. 11)<br />

Johnson, W.B., Texas A&M University, College Station, TX (Chs. 1, 19)<br />

Kisliakov, S., POMI, Sa<strong>in</strong>t Petersburg (Chs. 16, 20)<br />

Koldobsky, A., University <strong>of</strong> Missouri, Columbia, MO (Ch. 21)<br />

K6nig, H., University <strong>of</strong>Kiel, Kiel (Chs. 21, 22)<br />

L<strong>in</strong>denstrauss, J., The Hebrew University <strong>of</strong> Jerusalem, Jerusalem (Chs. 1, 15)<br />

Lomonosov, V. Kent State University, Kent, OH (Ch. 13)<br />

Milman, V.D., Tel-Aviv University, Tel-Aviv (Ch. 17)<br />

Odell, E.W., The University <strong>of</strong> Texas, Aust<strong>in</strong>, TX (Ch. 3)<br />

Phelps, R.R., University <strong>of</strong> Wash<strong>in</strong>gton, Seattle, WA (Ch. 15)<br />

Pietsch, A., Jena University, Jena (Ch. 11)<br />

Schachermayer, W., Vienna University <strong>of</strong> Technology, Vienna (Ch. 9)<br />

Schechtman, G., The Weizmann Institute <strong>of</strong> Science, Rehovot (Ch. 19)<br />

Szarek, S., Case Western Reserve University, Cleveland, OH<br />

and Universit~ Paris VI, Paris (Ch. 8)<br />

Wojtaszczyk, E, Warsaw University, Warsaw (Ch. 14)<br />

vii

CHAPTER 1<br />

<strong>Basic</strong> <strong>Concepts</strong> <strong>in</strong> <strong>the</strong> <strong>Geometry</strong> <strong>of</strong> <strong>Banach</strong> <strong>Spaces</strong><br />

William B. Johnson*<br />

Department <strong>of</strong> Ma<strong>the</strong>matics, Texas A &M University, College Station, TX 77843, USA<br />

E-mail: johnson @math.tamu.edu<br />

Joram L<strong>in</strong>denstrauss t<br />

Institute <strong>of</strong> Ma<strong>the</strong>matics, The Hebrew University <strong>of</strong> Jerusalem, Jerusalem, Israel<br />

E-mail: joram@math.huji.ac.il<br />

Contents<br />

1. Introduction .................................................. 3<br />

2. Notations and special <strong>Banach</strong> spaces ..................................... 3<br />

3. Bases ...................................................... 6<br />

4. Classical spaces ................................................ 13<br />

5. <strong>Banach</strong> lattices ................................................. 20<br />

6. <strong>Geometry</strong> <strong>of</strong> <strong>the</strong> norm ............................................. 30<br />

7. Analysis <strong>in</strong> <strong>Banach</strong> spaces .......................................... 35<br />

8. F<strong>in</strong>ite dimensional <strong>Banach</strong> spaces ...................................... 43<br />

9. Local structure <strong>of</strong> <strong>in</strong>f<strong>in</strong>ite dimensional spaces ................................ 53<br />

10. Some special classes <strong>of</strong> operators ...................................... 61<br />

11. Interpolation .................................................. 74<br />

12. List <strong>of</strong> symbols ................................................ 81<br />

References ..................................................... 83<br />

*Supported <strong>in</strong> part by NSF DMS-9623260 and DMS-9900185, Texas Advanced Research Program 010366-163,<br />

and US-Israel B<strong>in</strong>ational Science Foundation.<br />

+ Supported <strong>in</strong> part by US-Israel B<strong>in</strong>ational Science Foundation and by NSF DMS-9623260 as a participant <strong>in</strong><br />

<strong>the</strong> Workshop <strong>in</strong> L<strong>in</strong>ear Analysis & Probability at Texas A&M University.<br />

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 1<br />

Edited by William B. Johnson and Joram L<strong>in</strong>denstrauss<br />

9 2001 Elsevier Science B.V. All rights reserved

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces<br />

1. Introduction<br />

In this <strong>in</strong>troductory chapter we present results and concepts which are <strong>of</strong>ten used <strong>in</strong> <strong>Banach</strong><br />

space <strong>the</strong>ory and will be used <strong>in</strong> articles <strong>in</strong> this Handbook without fur<strong>the</strong>r reference. The<br />

material we treat, while familiar to experts <strong>in</strong> <strong>Banach</strong> space <strong>the</strong>ory, has not made its way<br />

<strong>in</strong>to <strong>in</strong>troductory courses <strong>in</strong> functional analysis. The ma<strong>in</strong> purpose <strong>of</strong> this article is to make<br />

<strong>the</strong> subsequent articles accessible to anyone whose background <strong>in</strong>cludes basic graduate<br />

courses <strong>in</strong> analysis and functional analysis. Each section <strong>of</strong> this article is devoted to one<br />

aspect <strong>of</strong> <strong>Banach</strong> space <strong>the</strong>ory.<br />

Although this article can <strong>in</strong> no way be considered as an <strong>in</strong>troductory course <strong>in</strong> <strong>Banach</strong><br />

space <strong>the</strong>ory, we do <strong>in</strong>clude <strong>in</strong>dications <strong>of</strong> pro<strong>of</strong> <strong>of</strong> some basic results <strong>in</strong> <strong>the</strong> hope that<br />

this will help <strong>the</strong> reader understand and get a feel<strong>in</strong>g for <strong>the</strong> various concepts which are<br />

discussed. We reference only some <strong>of</strong> <strong>the</strong> (mostly <strong>in</strong>troductory) books which treat <strong>the</strong> basic<br />

material we describe. Orig<strong>in</strong>al sources are referenced <strong>in</strong> <strong>the</strong>se books. We also mention<br />

some results which are not yet <strong>in</strong> elementary books on <strong>Banach</strong> spaces but which help to<br />

clarify <strong>the</strong> general picture. These generally were ei<strong>the</strong>r discovered recently or are more<br />

difficult than <strong>the</strong> rest <strong>of</strong> <strong>the</strong> material. In <strong>the</strong>se cases we refer to specific articles <strong>in</strong> this<br />

Handbook which treat <strong>the</strong> topic.<br />

In general, we do not attach names to <strong>the</strong>orems (except when experts generally refer to<br />

<strong>the</strong> <strong>the</strong>orem with a name attached, such as <strong>the</strong> Hahn-<strong>Banach</strong> <strong>the</strong>orem, <strong>the</strong> Kre<strong>in</strong>-Milman<br />

<strong>the</strong>orem, Rosenthal's s <strong>the</strong>orem .... ) or give any historical background. Instead we refer<br />

to <strong>the</strong> books <strong>in</strong> <strong>the</strong> references as well as <strong>the</strong> articles <strong>in</strong> this Handbook.<br />

2. Notations and special <strong>Banach</strong> spaces<br />

<strong>Banach</strong> spaces will have ei<strong>the</strong>r real or complex scalars. When <strong>the</strong> scalar field matters (for<br />

example, <strong>in</strong> results <strong>in</strong>volv<strong>in</strong>g spectral <strong>the</strong>ory or <strong>in</strong> <strong>the</strong>orems <strong>of</strong> an isometric nature or when<br />

analyticity plays a r61e), <strong>the</strong> scalar field is mentioned explicitly, but <strong>in</strong> <strong>the</strong> notation for<br />

special spaces <strong>the</strong> scalars are not specified.<br />

Operators between <strong>Banach</strong> spaces are bounded and l<strong>in</strong>ear. An <strong>in</strong>vertible operator T is<br />

called an isomorphism. Two norms on a vector space are called equivalent if <strong>the</strong> identity<br />

operator on X (with <strong>the</strong> two given norms) is an isomorphism. If IIT l] = 1 = I1T-1 ]], T is<br />

called an isometric isomorphism or simply an isometry and <strong>the</strong> doma<strong>in</strong> and range <strong>of</strong> T are<br />

said to be isometric. We write X ~ Y to denote that <strong>the</strong> spaces are isomorphic. To denote<br />

isometry we use <strong>the</strong> equal sign. An isomorphism from a <strong>Banach</strong> space onto itself is called<br />

an automorphism. A <strong>Banach</strong> space Y is said to be a quotient <strong>of</strong> <strong>the</strong> <strong>Banach</strong> space X if Y<br />

is isometric to X/Z for some closed subspace Z <strong>of</strong> X. By <strong>the</strong> open mapp<strong>in</strong>g <strong>the</strong>orem, Y is<br />

isomorphic to a quotient <strong>of</strong> X if <strong>the</strong>re is an operator from X onto Y.<br />

If X ~ Y, d(X, Y) denotes <strong>the</strong> <strong>Banach</strong>-Mazur distance between <strong>the</strong> spaces, def<strong>in</strong>ed<br />

to be <strong>the</strong> <strong>in</strong>fimum <strong>of</strong> IITII lIT -1 II as T ranges over all isomorphisms from X onto Y.<br />

So d(X, Y) = 1 if X and Y are isometric; <strong>the</strong> converse is true for f<strong>in</strong>ite dimensional<br />

spaces but not for <strong>in</strong>f<strong>in</strong>ite dimensional spaces. Note that <strong>the</strong> "triangle <strong>in</strong>equality" for <strong>the</strong><br />

<strong>Banach</strong>-Mazur distance is submultiplicative ra<strong>the</strong>r than subadditive; that is, d(X, Y)

- - Cont<strong>in</strong>uous<br />

- - C<br />

- The<br />

W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

A projection is an idempotent operator. A subspace Z <strong>of</strong> X is said to be complemented if<br />

<strong>the</strong>re is a projection from X onto Z. This is <strong>the</strong> case if and only if Z is closed and <strong>the</strong>re is a<br />

closed subspace W <strong>of</strong> X so that W A Z = {0} and X = W + Z; we <strong>the</strong>n write X = W 9 Z<br />

and say that X is <strong>the</strong> direct sum <strong>of</strong> W and Z. In this case W is isomorphic to X/Z by <strong>the</strong><br />

open mapp<strong>in</strong>g <strong>the</strong>orem.<br />

We regard X as a subspace <strong>of</strong> X** under <strong>the</strong> canonical embedd<strong>in</strong>g. Note that <strong>the</strong>re is<br />

always a projection <strong>of</strong> norm one from X*** onto X* (restrict <strong>the</strong> functionals on X** to X).<br />

An operator T :X ~ Y between <strong>Banach</strong> spaces is compact; respectively, weakly compact;<br />

if <strong>the</strong> image T Bx <strong>of</strong> <strong>the</strong> unit ball Bx <strong>of</strong> X has compact; respectively, weakly com-<br />

pact; closure <strong>in</strong> Y. If ei<strong>the</strong>r X or Y is reflexive, <strong>the</strong>n every operator from X to Y is weakly<br />

compact. The identity operator on X is compact if and only if X is locally compact if and<br />

only if X is f<strong>in</strong>ite dimensional.<br />

A sequence {Xn}n~ <strong>in</strong> a <strong>Banach</strong> space X is called weakly Cauchy provided {X*(Xn)}n~=l<br />

converges for every x* <strong>in</strong> X*. Identify<strong>in</strong>g a <strong>Banach</strong> space with a subspace <strong>of</strong> its bidual, and<br />

tak<strong>in</strong>g <strong>in</strong>to account that <strong>the</strong> uniform boundedness pr<strong>in</strong>ciple implies that a weakly Cauchy<br />

sequence is bounded, we see that {Xn }nO~ 1 is weakly Cauchy <strong>in</strong> X if and only if it converges<br />

weak* <strong>in</strong> X**. The space X is said to be weakly sequentially complete provided every<br />

weakly Cauchy sequence {Xn}neC__ 1 <strong>in</strong> X converges weakly, which is <strong>the</strong> same as say<strong>in</strong>g that<br />

<strong>the</strong> weak* limit <strong>of</strong> {Xn}n~176 1 <strong>in</strong> X** is <strong>in</strong> X itself.<br />

Here is a list <strong>of</strong> special classical <strong>Banach</strong> spaces and o<strong>the</strong>r objects. The elementary <strong>the</strong>ory<br />

<strong>of</strong> <strong>the</strong>se can be found <strong>in</strong> beg<strong>in</strong>n<strong>in</strong>g texts <strong>in</strong> real and functional analysis. A more detailed<br />

list <strong>of</strong> symbols, <strong>in</strong>clud<strong>in</strong>g some notation undef<strong>in</strong>ed <strong>in</strong> <strong>the</strong> text because we regard it as<br />

"standard", is conta<strong>in</strong>ed <strong>in</strong> Section 12.<br />

N<br />

R<br />

C<br />

T<br />

C(K;X)<br />

C(K)<br />

Lp(lZ)<br />

L~(#)<br />

Lp(T)<br />

~p(F)<br />

ep<br />

n<br />

,ep<br />

C<br />

co(F)<br />

-- The natural numbers.<br />

= The real numbers.<br />

= The complex numbers.<br />

= The unit circle <strong>in</strong> <strong>the</strong> complex plane.<br />

functions f on <strong>the</strong> (usually) compact Hausdorff space<br />

K tak<strong>in</strong>g values <strong>in</strong> <strong>the</strong> (usually) <strong>Banach</strong> space X, normed by Ilfll =<br />

suPtcg IIf(t)ll.<br />

(K; X) when X is <strong>the</strong> scalar field.<br />

-- The #-measurable functions f for which Ilfllp -- (f Ifl p d/z) 1/p < ~x~.<br />

Here 0 < p < cxz.<br />

#-measurable essentially bounded functions, with norm IIf I1~ --<br />

<strong>in</strong>fuz=0 sup Ifl~l. Lp(O, 1) -- Lp(#) when/z is Lebesgue measure on <strong>the</strong><br />

unit <strong>in</strong>terval.<br />

-- Lp (#) when # is normalized Lebesgue measure on <strong>the</strong> unit circle.<br />

- Lp(lZ) when # is count<strong>in</strong>g measure on <strong>the</strong> set F.<br />

- ~p(F) when F = N.<br />

--gp(/-') when F = {1, 2 ..... n}.<br />

= The subspace <strong>of</strong> eoc <strong>of</strong> scalar sequences which have a limit.<br />

- The closure <strong>in</strong> goc (F) <strong>of</strong> <strong>the</strong> scalar sequences which have f<strong>in</strong>ite support.

c0<br />

Bx<br />

Bx(x,r)<br />

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces<br />

= co(l-') when F = 1~.<br />

-- The closed unit ball <strong>of</strong> <strong>the</strong> <strong>Banach</strong> space X.<br />

= The closed ball <strong>of</strong> radius r with center x <strong>in</strong> <strong>the</strong> <strong>Banach</strong> space X; denoted<br />

also B(x, r) when X is understood.<br />

When 0 < p < 1, <strong>the</strong> space Lp(#) is not a <strong>Banach</strong> space except <strong>in</strong> <strong>the</strong> trivial cases that<br />

it is zero or one dimensional. The metric on L p(#), 0 < p < 1, is given by p(x, y) =<br />

IIx- yll p.<br />

If {X~}n~__l is a sequence <strong>of</strong> <strong>Banach</strong> spaces and 1

W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

product <strong>of</strong> <strong>the</strong> probability spaces (I-2n, I~'n) and for co = {con}nC~_ 1 <strong>in</strong> 1-2 set gn(co) = fn(con).<br />

In particular, it is possible to def<strong>in</strong>e a sequence <strong>of</strong> <strong>in</strong>dependent random variables hav<strong>in</strong>g<br />

standard Gaussian distribution on <strong>the</strong> <strong>in</strong>f<strong>in</strong>ite product (0, 1)N <strong>of</strong> (0, 1) (with <strong>the</strong> product <strong>of</strong><br />

Lebesgue measure). By <strong>the</strong> isomorphism <strong>the</strong>orem for separable measure algebras [ 18, p.<br />

399], (0, 1)r~ can be replaced by (0, 1) itself, but it is <strong>of</strong>ten more convenient to work on <strong>the</strong><br />

product space.<br />

The characteristic function <strong>of</strong> <strong>the</strong> random variable g is <strong>the</strong> function ~0 :R --+ C def<strong>in</strong>ed<br />

by ~0(t) = Ee itg. Useful algebraic identities <strong>in</strong>clude ~0(-t) = ~0(t); qgag+b(t ) -- e ibt~O(at);<br />

and, especially, qgf+g = qgf~Og, valid when f and g are <strong>in</strong>dependent. The characteristic<br />

function <strong>of</strong> a standard Gaussian random variable is e -t2/2.<br />

A key fact is that two random variables (possibly def<strong>in</strong>ed on different probability spaces)<br />

have <strong>the</strong> same distribution if and only if <strong>the</strong>y have <strong>the</strong> same characteristic function. This is<br />

proved by an <strong>in</strong>version formula [ 10, 2.3.a], ano<strong>the</strong>r simple consequence <strong>of</strong> which is that <strong>the</strong><br />

characteristic function <strong>of</strong> a random variable g is real valued if and only if g is symmetric;<br />

that is, g and -g have <strong>the</strong> same distribution. A (by no means immediate) consequence <strong>of</strong><br />

<strong>the</strong> <strong>in</strong>version formula [10, 2.7] is that for 0 < r < 2 <strong>the</strong> function e -Itlr is <strong>the</strong> characteristic<br />

function <strong>of</strong> a (necessarily symmetric) random variable, called a symmetric r-stable random<br />

variable. The tail distribution ~[Igl > t] <strong>of</strong> a symmetric r-stable random variable g is like<br />

t -r as t ~ c~, so that Ilgllp < ~ for p < r, but Ilgllr - cx~.<br />

Ano<strong>the</strong>r probabilistic notion that plays an important role <strong>in</strong> <strong>Banach</strong> space <strong>the</strong>ory is that<br />

<strong>of</strong> mart<strong>in</strong>gale. First we recall <strong>the</strong> notion <strong>of</strong> conditional expectation. Let ~ be a probability<br />

measure on a a-algebra/3. If ,4 is a sub cr-algebra <strong>of</strong> 13 and f is a ~-<strong>in</strong>tegrable function,<br />

<strong>the</strong>n by <strong>the</strong> Radon-Nikod3~m <strong>the</strong>orem <strong>the</strong>re is a A-measurable function g so that for each<br />

A <strong>in</strong> A, fA f dip -- fA g d~. The function g is called <strong>the</strong> conditional expectation <strong>of</strong> f given<br />

A and is sometimes denoted by E(f I A) Suppose now that {fn}~ is a sequence <strong>of</strong><br />

9<br />

F/=0<br />

random variables on <strong>the</strong> same probability space and/3n is <strong>the</strong> smallest cr-algebra for which<br />

fo, fl,..., fn are measurable. The sequence {fn }n~_0 is called a mart<strong>in</strong>gale provided that<br />

for each n, fn -- E(fn+l I/3n). The sequence {dn}n~=0 <strong>of</strong> differences; do = fo, dn - fn -<br />

fn-1; is <strong>the</strong>n called a mart<strong>in</strong>gale difference sequence. Notice that a sequence {dn}n~=0<br />

<strong>of</strong> <strong>in</strong>dependent random variables is a mart<strong>in</strong>gale difference sequence if and only if dn<br />

has mean zero for all n ~> 1. A simple but important example <strong>of</strong> a mart<strong>in</strong>gale difference<br />

sequence which is not a sequence <strong>of</strong> <strong>in</strong>dependent random variables is <strong>the</strong> Haar system,<br />

discussed <strong>in</strong> Section 4. One basic <strong>the</strong>orem about mart<strong>in</strong>gales, <strong>the</strong> mart<strong>in</strong>gale convergence<br />

<strong>the</strong>orem (see [10, 4.2.10]), states that every L1 bounded mart<strong>in</strong>gale converges a.e., which<br />

means that if {fn }n=0 is a mart<strong>in</strong>gale <strong>of</strong> P-measurable functions and SUPn Elfnl < c~, <strong>the</strong>n<br />

{fn }n~__0 converges a.e. Moreover, if <strong>the</strong> mart<strong>in</strong>gale is uniformly <strong>in</strong>tegrable (see Section 4,<br />

<strong>the</strong>n it also converges <strong>in</strong> L1 (~) (see [10, 4.5.3]).<br />

3. Bases<br />

Except<strong>in</strong>g [2] and [12], which are oriented to <strong>Banach</strong> space <strong>the</strong>ory, few <strong>in</strong>troductory texts<br />

<strong>in</strong> functional analysis treat Schauder bases. Never<strong>the</strong>less, bases are a very useful tool for<br />

<strong>in</strong>vestigat<strong>in</strong>g properties <strong>of</strong> <strong>Banach</strong> spaces.

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces<br />

We prove few statements <strong>in</strong> this section s<strong>in</strong>ce most <strong>of</strong> <strong>the</strong> results are proved <strong>in</strong> [2] and<br />

[12]. The book [14] <strong>of</strong>ten conta<strong>in</strong>s only sketches <strong>of</strong> pro<strong>of</strong>s. Chapter 2 <strong>of</strong> [21] conta<strong>in</strong>s<br />

enough details to be pleasant read<strong>in</strong>g for <strong>the</strong> mature student or experienced ma<strong>the</strong>matician<br />

who is not an expert <strong>in</strong> <strong>Banach</strong> space <strong>the</strong>ory. All <strong>of</strong> <strong>the</strong>se books conta<strong>in</strong> exercises. Many <strong>in</strong><br />

[21 ] are challeng<strong>in</strong>g even after peek<strong>in</strong>g at <strong>the</strong> h<strong>in</strong>ts. Exercises <strong>in</strong> [ 14], as <strong>in</strong> this <strong>in</strong>troductory<br />

article, are scattered throughout <strong>the</strong> text and are sometimes flagged by such expressions as<br />

"clearly", "hence", and <strong>the</strong> like.<br />

A Schauder basis or simply a basis for a <strong>Banach</strong> space X is a sequence {Xn}n~=l <strong>of</strong><br />

vectors <strong>in</strong> X such that every vector <strong>in</strong> X has a unique representation <strong>of</strong> <strong>the</strong> form ~ OlnX n<br />

with each C~n a scalar and where <strong>the</strong> sum is converges <strong>in</strong> <strong>the</strong> norm topology. The mapp<strong>in</strong>g<br />

x w-> Otn <strong>the</strong>n def<strong>in</strong>es for each n a l<strong>in</strong>ear functional x n on X. One checks that <strong>the</strong><br />

expression !xV 9 = SUPn IIE~=I xk (x)x~ll def<strong>in</strong>es a stronger complete norm on X, so that !.!<br />

and II, II are equivalent by <strong>the</strong> open mapp<strong>in</strong>g <strong>the</strong>orem. One deduces from this that <strong>the</strong><br />

biorthogonal functionals for a basis are necessarily cont<strong>in</strong>uous. Moreover, <strong>the</strong> biorthogonal<br />

functionals are a basic sequence <strong>in</strong> X*; that is, <strong>the</strong>y form a basis for <strong>the</strong>ir closed l<strong>in</strong>ear<br />

span. When it is useful to specify <strong>the</strong> biorthogonal functionals, we sometimes refer to <strong>the</strong><br />

"basis" {Xn, x n }n=l"<br />

A sequence {Xn}n~=l is called normalized if each vector Xn has norm one and {Xn}n~__l is<br />

called sem<strong>in</strong>ormalized if it is bounded and bounded away from zero. For most purposes it<br />

is sufficient to consider normalized or at least sem<strong>in</strong>ormalized basic sequences9<br />

The partial sum projections <strong>of</strong> a basis {Xn,'A n*l~ln__l ' def<strong>in</strong>ed for n = 1, 2,... by Pnx --<br />

Y~.<strong>in</strong>l x*(x)xi, are necessarily uniformly bounded and converge strongly to <strong>the</strong> identity;<br />

that is, Ilx - PnX II --+ 0 for each x <strong>in</strong> X. The supremum <strong>of</strong> <strong>the</strong> norms <strong>of</strong> <strong>the</strong>se partial sum<br />

projections is called <strong>the</strong> basis constant. This quantitative notion is <strong>of</strong> <strong>in</strong>terest also if we just<br />

consider f<strong>in</strong>ite basic sequences or bases for f<strong>in</strong>ite dimensional spaces. A sequence {x,},~__l<br />

<strong>of</strong> nonzero vectors is basic with basis constant at most C if and only if for all n < m (and<br />

all scalars O/i) <strong>the</strong> <strong>in</strong>equality II Y~<strong>in</strong>~__l OliXi ]l ~ C II Zi%l Oli Xi II holds. A block basis {yj }j= 1<br />

<strong>of</strong> <strong>the</strong> basis {x~}~__l is a sequence <strong>of</strong> nonzero vectors <strong>of</strong> <strong>the</strong> form yj<br />

m ~-'~nJ<br />

z--~k=nj+l<br />

+1<br />

~kxk for<br />

some sequence n l < n2 < .- .. The basis constant <strong>of</strong> a block basis <strong>of</strong> {Xn }n=~ 1 is no larger<br />

than <strong>the</strong> basis constant <strong>of</strong> {xn}~<br />

n=l"<br />

A basis is called monotone provided that its basis constant is one. One can change to an<br />

equivalent norm, Ill'Ill, <strong>in</strong> which <strong>the</strong> basis is monotone" just set ]llxlll- sup~ IlPnxll. In fact,<br />

if one def<strong>in</strong>es <strong>in</strong>stead Illxlll = SUPn

W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

you select Xn+l SO that maxx,cS [[Xn+l [[-1 [X*(Xn+l)[ is sufficiently small, <strong>the</strong> result<strong>in</strong>g sequence<br />

{Xn}n~__l will have basis constant less than e + I-In~__l Cn. This yields, for example,<br />

that every weakly null, non-norm null sequence has a basic subsequence.<br />

This last result can be improved substantially. Two basic sequences {Xn }n~__l and {Yn }n~=l<br />

are equivalent provided that <strong>the</strong> map Txn = Yn extends to an isomorphism from <strong>the</strong> closed<br />

span <strong>of</strong> {Xn}nC~=l onto <strong>the</strong> closed span <strong>of</strong> {Yn}n~ K-equivalent if [[TllllT-1[[ ~< K (so,<br />

strangely, {Xn}n~__l is 1-equivalent to {2Xn}n~__l, but usually basic sequences are normalized).<br />

The pr<strong>in</strong>ciple <strong>of</strong> small perturbations says that if {Xn}n~__l is a basic sequence <strong>in</strong><br />

X and IlXn - Yn II--+ 0 sufficiently quickly, <strong>the</strong>n {Yn}n~__l is a basic sequence which is<br />

equivalent to {Xn}n~=l. To prove this, let {Xn*}n~__l be Hahn-<strong>Banach</strong> extensions <strong>of</strong> <strong>the</strong> functionals<br />

biorthogonal to {Xn}n~__l to functionals <strong>in</strong> X* and def<strong>in</strong>e an operator S on X by<br />

Sx -- Y~x*(x)(yn -Xn), SO [[S[[ ~ E [[x*[[[[yn --Xn[[. If I]sll < 1, elementary considerations<br />

yield that I + S is an automorphism <strong>of</strong> X which maps Xn to Yn. This also yields<br />

that if { n}n=l is a basis <strong>of</strong> X, <strong>the</strong>n so is {Yn}n~_l. From this one sees that if {Xn}n~__l is<br />

a basis for X with biorthogonal functionals tx n *~ J n=l' if {Yn}n~=l is a sem<strong>in</strong>ormalized sequence<br />

<strong>in</strong> X, and if for each k, x k (Yn) -+ 0 as n --+ cx~, <strong>the</strong>n {y~}n~__l has a subsequence<br />

which is equivalent to some block basis {Zn}n~__l <strong>of</strong> {Xn}n~__l. From this it follows that every<br />

<strong>in</strong>f<strong>in</strong>ite dimensional subspace Y <strong>of</strong> a <strong>Banach</strong> space X with a basis {Xn}n~__l conta<strong>in</strong>s, for<br />

every K > 1, a normalized basic sequence {Zn}n~__l which is K-equivalent to a normalized<br />

block basis {Yn }n~--_ 1 <strong>of</strong> {Xn}~__1. Moreover, <strong>the</strong> small perturbation argument <strong>in</strong>dicated<br />

above shows that <strong>the</strong> generated isomorphism which maps {Yn}n~=l <strong>in</strong>to Y extends to an<br />

automorphism on X.<br />

To see how <strong>the</strong>se basic facts about bases might be used, suppose that T is a noncompact<br />

operator from a subspace X0 <strong>of</strong> a <strong>Banach</strong> space X with basis {Xn, x~ }n~__l <strong>in</strong>to a <strong>Banach</strong><br />

space Y with basis {Zn}n= 1. Then <strong>the</strong>re is a sequence {Yn}n~__l <strong>in</strong> <strong>the</strong> unit ball <strong>of</strong> X0 such<br />

that for some E > 0 and all n ~ m, IITyn - Tym I[ > ~. By pass<strong>in</strong>g to a subsequence <strong>of</strong><br />

differences <strong>of</strong> {Yn}n~=l it can be assumed that for each k, x k (Yn) --+ 0 as n --+ ~. In view<br />

<strong>of</strong> what was discussed <strong>in</strong> <strong>the</strong> previous paragraph, by pass<strong>in</strong>g to ano<strong>the</strong>r subsequence it can<br />

be assumed that {Yn}n~=l is an arbitrarily small perturbation <strong>of</strong> a block basis <strong>of</strong> {Xn}n~__l.<br />

Repeat<strong>in</strong>g <strong>the</strong> same argument for {Tyn}n~=l <strong>in</strong> Y and normaliz<strong>in</strong>g at <strong>the</strong> end, we conclude<br />

that if T is a noncompact operator from a subspace Xo <strong>of</strong> a <strong>Banach</strong> space X with basis<br />

{Xn}n~__l <strong>in</strong>to a <strong>Banach</strong> space Y with basis {Zn}n~=l, <strong>the</strong>n <strong>the</strong>re are automorphisms U on X<br />

and V on Y and a normalized block basis {Yn}n~__l <strong>of</strong> {Xn}n~ with Yn E U-1Xo and such<br />

that {VTUyn}n~__l is a sem<strong>in</strong>ormalized block basis <strong>of</strong> {Zn}n~__l.<br />

This last mentioned result gives considerable <strong>in</strong>formation <strong>in</strong> cases where all block bases<br />

<strong>of</strong> a basis can be characterized. The simplest examples <strong>of</strong> bases are provided by <strong>the</strong> unit<br />

vector basis {en}n~__l for s 1

We should mention that, <strong>in</strong> contrast to <strong>the</strong> case <strong>of</strong> gp and ~r, it generally is quite difficult<br />

to determ<strong>in</strong>e whe<strong>the</strong>r two spaces are isomorphic even when both are presented concretely<br />

as similar but different function spaces.<br />

Ano<strong>the</strong>r consequence <strong>of</strong> <strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong> small perturbations is that g l has <strong>the</strong> Schur<br />

property, which means that every weakly convergent sequence <strong>in</strong> ~ j is norm convergent.<br />

Indeed, o<strong>the</strong>rwise <strong>the</strong>re would be a weakly null sequence {Xn }n~=l <strong>in</strong> g l which is bounded<br />

away from zero. The sequence {Xn}n~=l is necessarily bounded, so <strong>the</strong> sequence {Xn}n~_l<br />

would have a subsequence {Xnk }~=l which is equivalent to a block basis <strong>of</strong> <strong>the</strong> unit vector<br />

basis <strong>of</strong> g 1 and thus equivalent to <strong>the</strong> unit vector basis <strong>of</strong> g 1. This is a contradiction because<br />

<strong>the</strong> unit vector basis <strong>of</strong> ~l is not weakly null. An easy formal consequence <strong>of</strong> <strong>the</strong> fact that<br />

g l has <strong>the</strong> Schur property is that every weakly Cauchy sequence <strong>in</strong> f j is norm convergent.<br />

In particular, el is weakly sequentially complete.<br />

The most natural basis for Lp(O, 1), 1 ~ p < oo, is <strong>the</strong> Haar system {hn}~_0, where<br />

h0 = 1 [0,1), and for n - 2 j 4- k with j = 0, 1 .... and k = 0, 1 ..... 2 j - 1,<br />

hn = l[k2-J,(2k+l)2-J-1 ) -- l[(2k+l)2-j-~,(k+l)2-j ).<br />

It is easy to check that if 1 ~< p ~< oo, <strong>the</strong>n for each n and all scalars Oti <strong>the</strong> <strong>in</strong>equality<br />

II~,i=o~ihillp n

10 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

A sequence which is an unconditional basis for its closed l<strong>in</strong>ear span is said to be unconditionally<br />

basic. S<strong>in</strong>ce a block basis <strong>of</strong> an unconditional basis is clearly unconditional (with<br />

unconditional constant no larger than that <strong>of</strong> <strong>the</strong> basis itself), <strong>the</strong> perturbation pr<strong>in</strong>ciple described<br />

above yields that every <strong>in</strong>f<strong>in</strong>ite dimensional subspace <strong>of</strong> a space with unconditional<br />

basis conta<strong>in</strong>s an unconditionally basic sequence.<br />

The unit vector bases for ep, 1 ~< p < cx~, and co are <strong>the</strong> simplest examples <strong>of</strong> unconditional<br />

bases. For 1 < p < c~, <strong>the</strong> Haar system forms an unconditional basis for L p (0, 1).<br />

The "modem" pro<strong>of</strong> <strong>of</strong> this proceeds via mart<strong>in</strong>gale <strong>the</strong>ory. The exact unconditional constant<br />

<strong>of</strong> <strong>the</strong> Haar system <strong>in</strong> Lp(O, 1) is computed <strong>in</strong> [23]. In Section 8 we po<strong>in</strong>t out that <strong>the</strong><br />

trigonometric system is an unconditional basis <strong>in</strong> Lp(7~) only for p -- 2.<br />

A basis {Xn}n~__ 1 for X is shr<strong>in</strong>k<strong>in</strong>g provided <strong>the</strong> l<strong>in</strong>ear span <strong>of</strong> <strong>the</strong> biorthogonal functionals<br />

{Xn}n= * ~ 1 is (norm) dense <strong>in</strong> X*, which is to say that {Xn*}n~__l is a basis for X*. Notice<br />

that a bounded shr<strong>in</strong>k<strong>in</strong>g basis necessarily converges weakly to zero. The basis {Xn}n~__l is<br />

boundedly complete provided that whenever <strong>the</strong> sequence {y~<strong>in</strong>__ 101i Xi }nC~_ 1 is bounded, <strong>the</strong>n<br />

it is convergent. If a basis is shr<strong>in</strong>k<strong>in</strong>g, <strong>the</strong>n its biorthogonal functionals are a boundedly<br />

complete basis for X*. Conversely, if <strong>the</strong> biorthogonal functionals form a boundedly complete<br />

basis for <strong>the</strong>ir closed l<strong>in</strong>ear span Y, <strong>the</strong>n <strong>the</strong> basis is shr<strong>in</strong>k<strong>in</strong>g (and hence Y = X*).<br />

Similarly, a basis is boundedly complete if and only if its biorthogonal functionals are a<br />

shr<strong>in</strong>k<strong>in</strong>g basis for <strong>the</strong>ir closed l<strong>in</strong>ear span. If {Xn}n~__l is a boundedly complete basis for<br />

X and Y is <strong>the</strong> normed closed l<strong>in</strong>ear span <strong>of</strong> <strong>the</strong> biorthogonal functionals {xn* }n=l' ~ <strong>the</strong><br />

shr<strong>in</strong>k<strong>in</strong>gness <strong>of</strong> {Xn*}n~__l implies that <strong>the</strong> natural evaluation mapp<strong>in</strong>g from X <strong>in</strong>to Y* is a<br />

surjective isomorphism (which is easily seen to be an isometry if <strong>the</strong> basis is monotone).<br />

Consequently, a space with a boundedly complete basis is isomorphic to a separable conjugate<br />

space. From <strong>the</strong>se facts it follows, <strong>in</strong> particular, that a <strong>Banach</strong> space X with a basis<br />

is reflexive if and only if some (or every) basis for X is both shr<strong>in</strong>k<strong>in</strong>g and boundedly<br />

complete.<br />

If X has an unconditional basis, <strong>the</strong>n some (or every) unconditional basis is boundedly<br />

complete if and only if X has no subspace isomorphic to co, while some (or every) unconditional<br />

basis is shr<strong>in</strong>k<strong>in</strong>g if and only if no (or no complemented) subspace <strong>of</strong> X is<br />

isomorphic to el. Consequently, a space with unconditional basis is reflexive if and only if<br />

no subspace is isomorphic to ei<strong>the</strong>r co or to ~1 if and only if its second dual is separable.<br />

In contrast, <strong>the</strong>re is a nonreflexive space which has both a shr<strong>in</strong>k<strong>in</strong>g basis and a boundedly<br />

complete basis which is <strong>of</strong> codimension one <strong>in</strong> its bidual, [ 14, 1.d.2].<br />

While most <strong>of</strong> <strong>the</strong> separable spaces encountered <strong>in</strong> classical analysis have bases, many<br />

do not have an unconditional basis; <strong>in</strong> particular, <strong>the</strong> spaces L1 (0, 1) and C[0, 1 ]. Indeed,<br />

co does not embed isomorphically <strong>in</strong>to L1 (0, 1) (for example, because L1 (0, 1) is weakly<br />

sequentially complete and co is not; see Section 4), so any unconditional basis for L1 (0, 1)<br />

would have to be boundedly complete and L 1(0, 1) would be isomorphic to a separable<br />

conjugate space. But L 1(0, 1) does not even embed isomorphically <strong>in</strong>to a separable conjugate<br />

space (s<strong>in</strong>ce if L 1(0, 1) C X* and for t E (0, 1) x~ is a weak* cluster po<strong>in</strong>t <strong>of</strong> <strong>the</strong><br />

sequence n -l l(t,t+l/n), <strong>the</strong>n it can be checked that <strong>the</strong>re is an uncountable subset S <strong>of</strong><br />

(0, 1) and 6 > 0 so that for all t :~ s <strong>in</strong> S, IIx? - x* II > 6). A more sophisticated (but actually<br />

technically easier) argument shows even that L1 (0, 1) does not embed isomorphically<br />

<strong>in</strong>to a space with unconditional basis; see [14, 1.d.1]. S<strong>in</strong>ce, as mentioned <strong>in</strong> Section 4,

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 11<br />

every separable space embeds isometrically <strong>in</strong>to C[0, 1 ], C[0, 1 ] also does not embed isomorphically<br />

<strong>in</strong>to a space with unconditional basis.<br />

It is a deep result that <strong>the</strong>re exist <strong>in</strong>f<strong>in</strong>ite dimensional spaces which do not conta<strong>in</strong> an<br />

unconditionally basic sequence (see [37]). The example uses <strong>Banach</strong> spaces whose norm<br />

is not def<strong>in</strong>ed explicitly by a formula but by an implicit procedure. This method <strong>of</strong> def<strong>in</strong><strong>in</strong>g<br />

a <strong>Banach</strong> space (or a norm) is very useful <strong>in</strong> many contexts.<br />

A basis {Xn}n~= 1 is called symmetric provided every permutation <strong>of</strong> {Xn}n~__ 1 is equivalent<br />

to {Xn}n~176. In particular, every permutation <strong>of</strong> {Xn}n~__l is a basis, so a symmetric basis is<br />

unconditional. A basis {Xn}n~__l is called subsymmetric provided it is unconditional and<br />

equivalent to each subsequence <strong>of</strong> itself. A symmetric basis is subsymmetric [14, 3.a.3].<br />

It is evident that if a basis is unconditional, symmetric, or subsymmetric, <strong>the</strong>n <strong>the</strong> same is<br />

true for <strong>the</strong> biorthogonal functionals (<strong>in</strong> <strong>the</strong>ir closed l<strong>in</strong>ear span).<br />

The unit vector bases for ~p, 1 ~< p < ec, and co are symmetric, while <strong>the</strong> L p-<br />

normalization <strong>of</strong> <strong>the</strong> Haar system is not a symmetric basis for L p (0, 1) if p 7~ 2. In fact,<br />

L p(O, 1) has no subsymmetric basis if p 7~ 2. We have already mentioned that L 1(0, 1)<br />

does not have even an unconditional basis. That L p(O, 1), 2 < p < ec, has no subsymmetric<br />

basis follows from <strong>the</strong> dichotomy pr<strong>in</strong>ciple for sequences <strong>in</strong> <strong>the</strong> space, which is<br />

discussed <strong>in</strong> Section 8. This pr<strong>in</strong>ciple says that if {xn}~~ 1 is a sem<strong>in</strong>ormalized basic sequence<br />

<strong>in</strong> Lp(O, 1), 2 < p < ec, <strong>the</strong>n {X~}nOC=l has a subsequence which is equivalent to<br />

<strong>the</strong> unit vector basis for ei<strong>the</strong>r s or g2. S<strong>in</strong>ce, as noted <strong>in</strong> Section 4, Lp(O, 1), p ~ 2, is<br />

not isomorphic to ei<strong>the</strong>r gp or ~2, Lp(O, 1), 2 < p < ec, has no subsymmetric basis. That<br />

L p(0, 1), 1 < p < 2, has no subsymmetric basis <strong>the</strong>n follows by duality.<br />

It is <strong>in</strong>terest<strong>in</strong>g that symmetric bases are never<strong>the</strong>less prevalent. For example, any space<br />

which has an unconditional basis is isomorphic to a complemented subspace <strong>of</strong> a space<br />

which has a symmetric basis (this will be discussed fur<strong>the</strong>r <strong>in</strong> Section 11). In fact, <strong>the</strong>re<br />

is even a space Y which has a symmetric basis and also an unconditional basis {Zn}n~<br />

which is universal for unconditional bases <strong>in</strong> <strong>the</strong> sense that every unconditional basis for<br />

any <strong>Banach</strong> space is equivalent to a subsequence <strong>of</strong> {Zn },c__ 1 ! The space Y and <strong>the</strong> universal<br />

unconditional basis is easy to construct, given <strong>the</strong> fact, to be proved <strong>in</strong> Section 4, that<br />

every separable <strong>Banach</strong> space is isometric to a subspace <strong>of</strong> C[0, 1]. Indeed, one takes a<br />

dense sequence {X~}neC=l <strong>of</strong> nonzero vectors <strong>in</strong> C[0, 1] and lets Y be <strong>the</strong> completion <strong>of</strong> <strong>the</strong><br />

f<strong>in</strong>itely supported scalar sequences under <strong>the</strong> norm II {an }~lll "- sup+ II Zn -+-anXn IIC[0,11.<br />

The unit vector basis <strong>in</strong> Y is a universal unconditional basis. See [14, p. 129] or [41] for<br />

fur<strong>the</strong>r discussion and for <strong>the</strong> pro<strong>of</strong> that <strong>the</strong> space Y has a symmetric basis.<br />

While unconditional bases, symmetric bases, and subsymmetric bases are all useful<br />

streng<strong>the</strong>n<strong>in</strong>gs <strong>of</strong> <strong>the</strong> notion <strong>of</strong> basis, <strong>the</strong>re are equally useful weaken<strong>in</strong>gs. A Schauder<br />

decomposition for a <strong>Banach</strong> space X is a sequence {X~ }n~__l <strong>of</strong> nonzero closed subspaces<br />

<strong>of</strong> X such that every vector x <strong>in</strong> <strong>the</strong> space has a unique representation <strong>of</strong> <strong>the</strong> form x -- ~ Xn<br />

with Xn ~ Xn. If each Xn is f<strong>in</strong>ite dimensional, <strong>the</strong> decomposition is called a f<strong>in</strong>ite dimensional<br />

decomposition, or simply an FDD. A basis can be regarded as an FDD { Xn }~ 1 such<br />

that every space Xn is one dimensional. As <strong>in</strong> <strong>the</strong> case <strong>of</strong> bases, a Schauder decomposition<br />

{Xn}~~1761 for a <strong>Banach</strong> space X determ<strong>in</strong>es a sequence {Pn }~~176 1 (called <strong>the</strong> partial sum<br />

projections <strong>of</strong> <strong>the</strong> decomposition) <strong>of</strong> commut<strong>in</strong>g projections with <strong>in</strong>creas<strong>in</strong>g ranges which<br />

converge strongly to <strong>the</strong> identity operator on X; namely, for x = y~ Xn with x~ E Xn and<br />

n - 1, 2 ..... Pnx - ~-~<strong>in</strong>l xi. Conversely, if {Pn}n~__l is a sequence <strong>of</strong> commut<strong>in</strong>g projec-

12 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

tions with <strong>in</strong>creas<strong>in</strong>g ranges which converge strongly to <strong>the</strong> identity operator on X, <strong>the</strong>n<br />

{(Pn - Pn-1)X}n~=l (P0 "-- 0) is a Schauder decomposition for X for which {Pn }n~_-i is <strong>the</strong><br />

sequence <strong>of</strong> partial sum projections. The biorthogonal functionals for a basis are replaced<br />

<strong>in</strong> <strong>the</strong> case <strong>of</strong> a Schauder decomposition by <strong>the</strong> sequence {(P* - Pn*l)X* }n~=l <strong>of</strong> (even<br />

weak*) closed subspaces <strong>of</strong> X*, which form a Schauder decomposition for <strong>the</strong>ir closed<br />

span. The supremum <strong>of</strong> <strong>the</strong> norms <strong>of</strong> <strong>the</strong> partial sum projections determ<strong>in</strong>ed by a Schauder<br />

decomposition is f<strong>in</strong>ite and is called <strong>the</strong> decomposition constant <strong>of</strong> <strong>the</strong> decomposition, and<br />

<strong>the</strong> Schauder decomposition is called monotone if its decomposition constant is one.<br />

The def<strong>in</strong>itions <strong>of</strong> unconditional, shr<strong>in</strong>k<strong>in</strong>g, and boundedly complete bases generalize<br />

immediately to Schauder decompositions. The structure <strong>the</strong>ory for bases goes over with<br />

little difficulty to FDD's. More importantly, <strong>the</strong>re are useful concepts <strong>in</strong>volv<strong>in</strong>g FDD's<br />

which are not merely generalizations <strong>of</strong> notions about bases. For example, a block<strong>in</strong>g <strong>of</strong><br />

a Schauder decomposition {Xn}n~=l is a sequence <strong>of</strong> <strong>the</strong> form {Xnk "-[-''" + Xnk+l-1}L1<br />

with 1 = n 1 < n2 < .... Any block<strong>in</strong>g <strong>of</strong> a basis (that is, <strong>of</strong> a Schauder decomposition <strong>in</strong>to<br />

one dimensional spaces) produces an FDD which is not a basis. While bases and block<br />

bases are f<strong>in</strong>e <strong>in</strong> situations where passage to subsequences or subspaces <strong>in</strong>volves no loss,<br />

block<strong>in</strong>gs <strong>of</strong> FDD's sometime provide <strong>the</strong> proper framework for <strong>in</strong>vestigations where it<br />

is important to ma<strong>in</strong>ta<strong>in</strong> global control. For example, it was mentioned above that every<br />

normalized basic sequence <strong>in</strong> ep, 1 < p < cx~, has a subsequence equivalent to <strong>the</strong> unit<br />

vector basis <strong>of</strong> ep, but one can prove that every FDD for a subspace <strong>of</strong> gp, 1 < p < ~, has<br />

a block<strong>in</strong>g which is an ep decomposition [14, 2.d.1]. (A Schauder decomposition {Xn}n~__l<br />

is said to be an g.p decomposition provided that for Xn ~ Xn, <strong>the</strong> series y~ Xn converges if<br />

and only if y~ [[Xn[Ip < oo.)<br />

A separable <strong>Banach</strong> space is said to have <strong>the</strong> bounded approximation property or simply<br />

BAP provided <strong>the</strong>re is a sequence { Tn }n~__l <strong>of</strong> f<strong>in</strong>ite rank operators on X which converges<br />

strongly to <strong>the</strong> identity. If <strong>the</strong> operators can be chosen to commute, <strong>the</strong> space has <strong>the</strong> commut<strong>in</strong>g<br />

bounded approximation property (CBAP). In ei<strong>the</strong>r case <strong>the</strong> supremum <strong>of</strong> [[ Tn [[ is<br />

f<strong>in</strong>ite; if at most )~, we say that <strong>the</strong> space has <strong>the</strong> )~-BAP or )~-CBAP, or when )~ = 1 <strong>the</strong> metric<br />

approximation property (MAP) or commut<strong>in</strong>g metric approximation property (CMAP).<br />

F<strong>in</strong>ally, A <strong>Banach</strong> space X has <strong>the</strong> approximation property or simply AP provided that for<br />

every compact subset K <strong>of</strong> X and E > 0, <strong>the</strong>re is a f<strong>in</strong>ite rank operator T on X such that<br />

[Ix - Tx l[ < ~ for every x ~ K. If a space X has <strong>the</strong> AP, <strong>the</strong>n every compact operator S<br />

<strong>in</strong>to X is <strong>the</strong> norm limit <strong>of</strong> a sequence <strong>of</strong> f<strong>in</strong>ite rank operators. Indeed, take for n = 1, 2 ....<br />

a f<strong>in</strong>ite rank operator Tn on X so that for each x <strong>in</strong> <strong>the</strong> image under <strong>of</strong> <strong>the</strong> unit ball <strong>of</strong> <strong>the</strong><br />

doma<strong>in</strong> <strong>of</strong> S <strong>the</strong> <strong>in</strong>equality IIx - Tnxll

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 13<br />

conditions leads to nice <strong>the</strong>orems which do not <strong>in</strong>volve any notion <strong>of</strong> approximation; <strong>in</strong><br />

particular, isomorphic characterizations <strong>of</strong> Hilbert space (see [35]).<br />

The existence <strong>of</strong> spaces fail<strong>in</strong>g <strong>the</strong> AP as well as o<strong>the</strong>r considerations lead to <strong>the</strong> study<br />

<strong>of</strong> o<strong>the</strong>r basis-like structures <strong>in</strong> <strong>Banach</strong> spaces. Probably <strong>the</strong> most useful <strong>of</strong> <strong>the</strong>se is that <strong>of</strong><br />

Markuschevich basis. A Markuschevich basis for a <strong>Banach</strong> space X is a biorthogonal system<br />

{x• x• *}• for which {x• }y~r is fundamental (that is, <strong>the</strong> span <strong>of</strong> <strong>the</strong> x• is dense<br />

<strong>in</strong> X) and {X•215 * is total; that is, <strong>the</strong> x• *'s separate <strong>the</strong> po<strong>in</strong>ts <strong>of</strong> X. The trigonometric<br />

system <strong>in</strong> L I(~') or C(~') is a natural example <strong>of</strong> a Markuschevich basis which is not a<br />

(Schauder) basis <strong>in</strong> any order (but this last statement is not easy to verify). Markuschevich<br />

bases do not always exist <strong>in</strong> <strong>the</strong> nonseparable sett<strong>in</strong>g, but are a useful tool for <strong>in</strong>vestigat<strong>in</strong>g<br />

nonseparable spaces <strong>in</strong> situations where <strong>the</strong>y exist (see [44]). It is easy to see that<br />

any separable space X admits a Markuschevich basis. One starts with l<strong>in</strong>early <strong>in</strong>dependent<br />

sequences {Yn}nCX~ 1 <strong>in</strong> X and {'Yn*'CX~ln--I <strong>in</strong> X* with {Yn}n~ 1_ fundamental and {yn*}n~_l_ total<br />

and applies a Gram-Schmidt type procedure to biorthogonalize <strong>the</strong> sequences (alternat<strong>in</strong>g<br />

between work<strong>in</strong>g <strong>in</strong> X and X*) to produce a Markuschevich basis {Xn,X*}n~__l so that<br />

span{Xn}n~=l -- span{yn}n~_l _ and span{xn*}n~__l = s-an'- p [.Ynln= *~ 1 (see [14, 1.f.3]). A deeper<br />

fact [14, 1.f.4] is that a separable space conta<strong>in</strong>s a Markuschevich basis {Xn,Xn}n: * ~ 1 for<br />

which SUPn IIxn II IIxn*ll ~< 20, and it is known that twenty can be replaced by any number<br />

larger than one. We shall see <strong>in</strong> Section 8 that a f<strong>in</strong>ite dimensional space has a basis<br />

, N<br />

{Xn, x n }n=l for which IIxn II IIx* II - 1 for each n, but it is an open problem whe<strong>the</strong>r every<br />

separable space has a Markuschevich basis with this property. One separable <strong>the</strong>orem <strong>in</strong><br />

which Markuschevich bases are used but do not appear <strong>in</strong> <strong>the</strong> statement is that every separable<br />

space X has a subspace Y such that both Y and X~ Y have an FDD [ 14, 1.g.2]. Incidentally,<br />

it is open whe<strong>the</strong>r <strong>the</strong> conclusion can be improved to "both Y and X~ Y have a basis".<br />

4. Classical spaces<br />

In this section we present some basic facts concern<strong>in</strong>g <strong>the</strong> structure <strong>of</strong> <strong>the</strong> classical spaces<br />

C(K) and Lp(#), <strong>the</strong>ir classification, and <strong>the</strong> relations among <strong>the</strong> spaces. Most books<br />

on real analysis, such as [ 18], conta<strong>in</strong> some elementary results about <strong>the</strong> structure <strong>of</strong> <strong>the</strong>se<br />

spaces. Beyond <strong>the</strong> most basic material, [21 ] is accessible to students and has <strong>the</strong> additional<br />

advantage that some <strong>of</strong> <strong>the</strong> relations between <strong>the</strong>se spaces and o<strong>the</strong>r spaces encountered <strong>in</strong><br />

analysis are touched on. The more narrowly focused book [7] is also directed at students,<br />

conta<strong>in</strong>s several structural results omitted from [21], and <strong>the</strong> author's <strong>of</strong>f-<strong>the</strong>-wall style<br />

makes <strong>the</strong> book fun to read.<br />

A normalized sequence {Xn}n~=l <strong>of</strong> disjo<strong>in</strong>tly supported functions <strong>in</strong> Lp(#), 1 0 <strong>the</strong>re is a subspace Z <strong>of</strong> Y with<br />

d (Z, X) < 1 § ~ and so that <strong>the</strong>re is a projection <strong>of</strong> norm less than 1 + ~ from X onto Z.

14 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

Although general subspaces <strong>of</strong> X, X = s for 1 ~ p 7~ 2 < cc or X = co can be bad <strong>in</strong><br />

that <strong>the</strong>y can fail <strong>the</strong> approximation property and <strong>the</strong>re are many <strong>of</strong> <strong>the</strong>m, <strong>in</strong> a sense that is<br />

made precise <strong>in</strong> [24], <strong>the</strong> isomorphism <strong>the</strong>ory <strong>of</strong> complemented subspaces <strong>of</strong> X is simple:<br />

An <strong>in</strong>f<strong>in</strong>ite dimensional complemented subspace Y <strong>of</strong> X, X = g.p for 1

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 15<br />

Next, if Lp(l z) is separable with # a purely nonatomic probability measure, <strong>the</strong>n <strong>the</strong><br />

isomorphism <strong>the</strong>orem for separable measure algebras [18, p. 399] yields that Lp(tX) is<br />

isometric to L p(O, 1). On <strong>the</strong> o<strong>the</strong>r hand, if <strong>the</strong> measure # is purely atomic with atoms<br />

{A• <strong>the</strong>n <strong>the</strong> function which maps for each y <strong>the</strong> unit vector ey to lZ(Ay) -1/p times<br />

<strong>the</strong> <strong>in</strong>dicator function <strong>of</strong> A• extends to an isometry from g.p(F) onto Lp(lZ). Now if x<br />

and y are <strong>in</strong> Lp(#), 1 ~ p ~ 2 < oc, <strong>the</strong>n x and y are disjo<strong>in</strong>tly supported if and only if<br />

Ilx + yll p = IlXllp + Ilyll p [18, p. 416]. This implies that an isometry from one Lp space<br />

onto ano<strong>the</strong>r preserves disjo<strong>in</strong>tness and <strong>the</strong>refore also atoms. All <strong>the</strong>se remarks comb<strong>in</strong>e to<br />

show that g.np, ~p, Lp(O, 1), s @p Lp(O, 1), ~pn @p Lp(O, 1), n -- 1, 2 .... , is a complete<br />

list<strong>in</strong>g, up to isometry, <strong>of</strong> <strong>the</strong> separable Lp(#) spaces when 1 ~< p 7~ 2 < co, and <strong>the</strong>se are<br />

all mutually nonisometric. Of course, <strong>in</strong> <strong>the</strong> Hilbertian case p - 2, s n -- 1, 2 .... ; ~2 is<br />

<strong>the</strong> appropriate list<strong>in</strong>g.<br />

The decomposition method <strong>the</strong>n yields that s s L p(O, 1) is a complete list<strong>in</strong>g, up<br />

to isomorphism, <strong>of</strong> <strong>the</strong> separable L p spaces. Later <strong>in</strong> this section it is noted that s is not<br />

isomorphic to L p (0, 1) when 1 ~< p 7~ 2 < ec, so this is a list<strong>in</strong>g <strong>of</strong> nonisomorphic spaces<br />

for <strong>the</strong>se values <strong>of</strong> p.<br />

In order to study <strong>the</strong> structure <strong>of</strong> subspaces <strong>of</strong> L p (0, 1) as well as to <strong>in</strong>vestigate many<br />

o<strong>the</strong>r questions about L p for one fixed value <strong>of</strong> p, it is convenient to consider <strong>the</strong> scale<br />

<strong>of</strong> L p spaces as p varies. One reason for this is that (as we shall see) <strong>the</strong>re are spaces <strong>of</strong><br />

functions X on (0, 1) on which <strong>the</strong> norm II" II p, is equivalent to II" II P2 with Pl < P2. Note<br />

that when this occurs, s<strong>in</strong>ce I1" II p ~< I1" IIr when p ~< r, all <strong>the</strong> norms I1" II p are equivalent<br />

on X for pl ~< p ~< p2. In fact, <strong>the</strong> extrapolation pr<strong>in</strong>ciple says that <strong>the</strong>n all <strong>the</strong> norms I1" II p<br />

are equivalent on X for p ~< p2. Indeed, suppose C is a constant so that IIx II p2 ~< C IIx II pl<br />

for x E X, 0 < p < pl, and 0 < )~ < 1 is def<strong>in</strong>ed by <strong>the</strong> formula pl = )~p + (1 - )Qp2.<br />

Then, by H61der's <strong>in</strong>equality, for x E X we have<br />

IIx II p~ ~ IIx II ~p IIx II (1-~)p= ~ C 1-~ IIx II p~l-& IIx II~p<br />

and hence C 1-1/z IIx Ilpl ~ IIx lip ~ IIx IIp~.<br />

The preced<strong>in</strong>g yields particularly good <strong>in</strong>formation <strong>in</strong> case 2 < p and X is a closed<br />

subspace <strong>of</strong> L p(O, 1) which is closed <strong>in</strong> Lr (0, 1) for some r ~: p. By <strong>the</strong> open mapp<strong>in</strong>g<br />

<strong>the</strong>orem, <strong>the</strong> I1" II p and I1" II r norms are equivalent on X, and hence so is <strong>the</strong> I1" 112 norm. This<br />

means that X must be isomorphic to a Hilbert space. Moreover, <strong>the</strong> orthogonal projection<br />

P from L2 (0, 1) onto X <strong>in</strong>duces a bounded l<strong>in</strong>ear projection/3 from L p (0, 1) onto X. The<br />

extrapolation pr<strong>in</strong>ciple <strong>the</strong>n says that I1" lip*, 1/p + I/p* = 1, and I1" 112 are equivalent on<br />

X. The orthogonal projection onto X extends to <strong>the</strong> bounded projection 15, from Lp, (0, 1)<br />

onto X.<br />

Suppose that, on <strong>the</strong> o<strong>the</strong>r hand, X is a closed subspace <strong>of</strong> L p (0, 1), 1 ~< p < ~, and for<br />

some (or, equivalently, every) 0 < r < p <strong>the</strong> I1" II p and I1" Ilr norms are not equivalent on<br />

X. H61der's <strong>in</strong>equality <strong>the</strong>n yields that for any M < cx~ <strong>the</strong> <strong>in</strong>fimum <strong>of</strong> Ilxl[ixl

16 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

<strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong> small perturbations, that X conta<strong>in</strong>s a subspace which is isomorphic to s<br />

and complemented <strong>in</strong> L p (0, 1).<br />

These observations yield a dichotomy pr<strong>in</strong>ciple for subspaces <strong>of</strong> L p(O, 1), 2 < p <<br />

cx~ [21, III.A.4]. Let X be a closed subspace <strong>of</strong> L p(O, 1), 2 < p < oe. Then ei<strong>the</strong>r X<br />

is isomorphic to a Hilbert space and complemented <strong>in</strong> L p(O, 1) or, for each C > 1, X<br />

conta<strong>in</strong>s a subspace C-isomorphic to s and C-complemented <strong>in</strong> L p (0, 1). This is stated<br />

for Lp(O, 1), but is valid for general Lp(#) by <strong>the</strong> isometric classification <strong>of</strong> Lp spaces<br />

discussed earlier. Notice also that <strong>the</strong> comments <strong>in</strong> Section 3 about s now yield that if s<br />

embeds isomorphically <strong>in</strong>to L p(#), 0 < r < cx~ and 2 < p < oo, <strong>the</strong>n ei<strong>the</strong>r r = p or<br />

1~ --- 2.<br />

The natural examples <strong>of</strong> <strong>in</strong>f<strong>in</strong>ite dimensional function spaces which are closed <strong>in</strong> L p<br />

for more than one value <strong>of</strong> p come from probability <strong>the</strong>ory. Suppose that {gn}n~=l is a<br />

sequence <strong>of</strong> <strong>in</strong>dependent standard Gaussian random variables, discussed <strong>in</strong> Section 2. From<br />

<strong>the</strong> form <strong>of</strong> <strong>the</strong> distribution <strong>of</strong> g it is evident that ]lg]lp is f<strong>in</strong>ite for all f<strong>in</strong>ite p and {gn}n~=l<br />

is an orthonormal sequence <strong>in</strong> L2 (0, 1). A characteristic function argument yields that if<br />

n 2 n<br />

Y~=l [otk] -- 1, <strong>the</strong>n Y~k= 1 ctk gk is aga<strong>in</strong> a standard Gaussian and hence has <strong>the</strong> same<br />

norm <strong>in</strong> Lp(O, 1) as g. This means that for all 0 < p < oo, <strong>the</strong> mapp<strong>in</strong>g en ~ ][g][plgn,<br />

n = 1, 2 ..... extends to an isometry from s onto a subspace X <strong>of</strong> L p (0, 1) where X does<br />

not even depend on p, and <strong>the</strong> orthogonal projection onto X def<strong>in</strong>es a bounded projection<br />

from L p (0, 1) onto X if 1 < p < oe (<strong>in</strong> Section 10 we expla<strong>in</strong> why no isomorph <strong>of</strong> s<br />

<strong>in</strong> L 1(0, 1) or L~ (0, 1) can be complemented). In particular, <strong>the</strong> dichotomy pr<strong>in</strong>ciple for<br />

subspaces <strong>of</strong> L p, 2 < p < oo, is not a monochotomy pr<strong>in</strong>ciple. This also implies, <strong>in</strong> view<br />

<strong>of</strong> <strong>the</strong> structure <strong>the</strong>ory <strong>of</strong> s discussed earlier, that s and L p (0, 1) are not isomorphic for<br />

0 < p :/: 2 < o~.<br />

A second natural example from probability <strong>the</strong>ory <strong>of</strong> a subspace <strong>of</strong> L p, 0 < p < o0,<br />

which is isomorphic (but not isometric for p # 2) to s comes from consider<strong>in</strong>g a sequence<br />

{en}n~=l <strong>of</strong> <strong>in</strong>dependent random variables each tak<strong>in</strong>g on each <strong>of</strong> <strong>the</strong> values 1 and -1 with<br />

probability 1/2. Such a sequence {en}n~__l is called a Rademacher sequence. Classically<br />

<strong>the</strong> Rademacherfunctions {rn }n~__l are <strong>the</strong> concrete realization <strong>of</strong> such a sequence on [0, 1 ]<br />

m V ~2n - 1<br />

def<strong>in</strong>ed by rn " z_,~=o hz,+k, where {hn}n~__l is <strong>the</strong> Haar system. Kh<strong>in</strong>tch<strong>in</strong>e's <strong>in</strong>equality<br />

says that a Rademacher sequence is equivalent, <strong>in</strong> <strong>the</strong> Lp norm, 0 < p < o~, to <strong>the</strong> unit<br />

vector basis <strong>of</strong> s for future reference we write <strong>the</strong> <strong>in</strong>equality explicitly with <strong>the</strong> best<br />

constants labeled Ap and Bp:<br />

S<strong>in</strong>ce a Rademacher sequence is orthonormal,<br />

llz .ll<br />

it follows that Ap =- 1 for p ~> 2 and Bp = 1 for p ~< 2, and <strong>the</strong> exact values <strong>of</strong> Ap and Bp<br />

are known. The most elementary pro<strong>of</strong> <strong>of</strong> Kh<strong>in</strong>tch<strong>in</strong>e's <strong>in</strong>equality proceeds by check<strong>in</strong>g<br />

<strong>the</strong> right <strong>in</strong>equality <strong>in</strong> (1) for p an even <strong>in</strong>teger. This clearly gives <strong>the</strong> result for all 2 ~<<br />

p < oo and one <strong>the</strong>n obta<strong>in</strong>s <strong>the</strong> result for 0 < p < 2 from <strong>the</strong> extrapolation pr<strong>in</strong>ciple. The

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 17<br />

less computational modern pro<strong>of</strong> <strong>of</strong> Kh<strong>in</strong>tch<strong>in</strong>e's <strong>in</strong>equality gives a vector valued version<br />

<strong>of</strong> Kh<strong>in</strong>tch<strong>in</strong>e's <strong>in</strong>equality called <strong>the</strong> Kahane-Kh<strong>in</strong>tch<strong>in</strong>e <strong>in</strong>equality. This will be done <strong>in</strong><br />

Section 8.<br />

The existence <strong>of</strong> an r-stable variable g (see Section 2) for 0 < r < 2 shows that <strong>the</strong> subspace<br />

structure <strong>of</strong> L p (0, 1), 0 < p < 2, is much more complicated than that <strong>of</strong> L p (0, 1),<br />

2 < p < cx~. Indeed, just as <strong>in</strong> <strong>the</strong> Gaussian case, <strong>the</strong>re exists for 0 < r < 2 a sequence<br />

{gn}n~l <strong>of</strong> symmetric r-stable random variables def<strong>in</strong>ed on (0, 1). If 0 < p < r, <strong>the</strong>n <strong>the</strong>se<br />

random variables are <strong>in</strong> Lp(O, 1). Now if ~k--1 n I~l r 1, <strong>the</strong>n ~=l otkgk is aga<strong>in</strong> symmetric<br />

r-stable and hence has <strong>the</strong> same norm <strong>in</strong> L p(O, 1) as g. This means that for all<br />

0 < p < r, <strong>the</strong> mapp<strong>in</strong>g en ~ Ilgllp<br />

-1<br />

gn, n -- 1, 2 ..... extends to an isometry from er <strong>in</strong>to<br />

L p (0, 1). In Section 9 we expla<strong>in</strong> how to derive from this <strong>the</strong> fact that for 1 ~< p < r < 2,<br />

Lr (0, 1) embeds isometrically <strong>in</strong>to L p(O, 1). It turns out that this covers all <strong>the</strong> cases <strong>in</strong><br />

which er embeds isomorphically <strong>in</strong>to Lp(iZ ) with p and r f<strong>in</strong>ite. The rema<strong>in</strong><strong>in</strong>g cases are<br />

discussed <strong>in</strong> Section 8.<br />

In <strong>the</strong> reflexive range 1 < p < cx~, <strong>in</strong>formation about subspaces <strong>of</strong> Lp,(#), 1/p +<br />

1/p* - 1, given above gives <strong>in</strong>formation on quotients <strong>of</strong> Lp(#). It turns out that <strong>the</strong> case<br />

p = 1 is quite different: Every separable <strong>Banach</strong> space X is isometric to a quotient <strong>of</strong> el.<br />

Indeed, if {Xn}n~=l is dense <strong>in</strong> <strong>the</strong> unit ball <strong>of</strong> X, <strong>the</strong> l<strong>in</strong>ear extension <strong>of</strong> <strong>the</strong> map en ~ Xn<br />

(where {en}~--1 is <strong>the</strong> unit vector basis for el) maps <strong>the</strong> unit ball <strong>of</strong> el onto a dense subset<br />

<strong>of</strong> <strong>the</strong> unit ball <strong>of</strong> X and hence extends to a quotient mapp<strong>in</strong>g from el onto X. Similarly,<br />

if <strong>the</strong> <strong>Banach</strong> space X has density character x, <strong>the</strong>n X is isometric to a quotient <strong>of</strong> el (F)<br />

when F has card<strong>in</strong>ality x.<br />

Ano<strong>the</strong>r useful and <strong>in</strong>terest<strong>in</strong>g property <strong>of</strong> el (which also holds for el(F) for any set<br />

F) is <strong>the</strong> lift<strong>in</strong>g property: If T is an operator from a <strong>Banach</strong> space X onto el <strong>the</strong>n <strong>the</strong>re<br />

is a lift<strong>in</strong>g S <strong>of</strong> T; that is, an operator S :el --+ X for which T S = Ie~. Indeed, by <strong>the</strong> open<br />

mapp<strong>in</strong>g <strong>the</strong>orem <strong>the</strong>re are Xn <strong>in</strong> X with Txn = en and )~ :=- supllxn II < cx~. The mapp<strong>in</strong>g<br />

en ~ Xn <strong>the</strong>n extends to an operator S:e 1 ---+ X with [[Sll - ;~ satisfy<strong>in</strong>g TS = le~.<br />

As noted <strong>in</strong> Section 3, e 1 has <strong>the</strong> Schur property; that is, weakly convergent sequences <strong>in</strong><br />

el are norm convergent. An immediate consequence is that every weakly compact subset<br />

<strong>of</strong> el is norm compact. The weakly compact subsets <strong>of</strong> L 1(0, 1) are more complicated<br />

s<strong>in</strong>ce L 1(0, 1) conta<strong>in</strong>s <strong>in</strong>f<strong>in</strong>ite dimensional reflexive subspaces. There is however a nice<br />

characterization <strong>of</strong> subsets <strong>of</strong> L1 (/z) which have weakly compact closure when /z is a<br />

f<strong>in</strong>ite measure. First, if X is any <strong>Banach</strong> space and W is a subset <strong>of</strong> X such that for each<br />

E > 0 <strong>the</strong>re exists a weakly compact set S so that W C S + EBx, <strong>the</strong>n W has weakly<br />

compact closure (use <strong>the</strong> fact that a bounded subset <strong>of</strong> X is weakly compact if its weak*<br />

closure <strong>in</strong> X** is a subset <strong>of</strong> X). Next, given W C L1 (#) with # a f<strong>in</strong>ite measure, set<br />

for k E N, ak- ak(W) "--sup{llxllxl~klll" x ~ W}. Clearly {ak}~= 1 decreases to some<br />

a = a(W) ~ O. The set W is called uniformly <strong>in</strong>tegrable if a = 0. If a = 0 <strong>the</strong>n W has<br />

weakly compact closure because W is a subset <strong>of</strong> kBL~(~) + akBLl(~) and BL~(u) is<br />

weakly compact <strong>in</strong> L1 (/z).<br />

If a(W) > 0, <strong>the</strong> set W does not have weakly compact closure and <strong>in</strong> fact even conta<strong>in</strong>s<br />

a sequence equivalent to <strong>the</strong> unit vector basis <strong>of</strong> el. To see this, def<strong>in</strong>e two fur<strong>the</strong>r<br />

numerical parameters for a subset W <strong>of</strong> L 1 (#); namely, b(W) := sup limn~ { Ilxn 1 An II1 },<br />

where <strong>the</strong> supremum is over all sequences {Xn}n~__l <strong>in</strong> W and {An}~=l <strong>of</strong> measurable sets<br />

with #An --+ 0; and c(W), def<strong>in</strong>ed <strong>the</strong> same way as b(W) except that <strong>the</strong> supremum is

18 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

over all sequences <strong>of</strong> disjo<strong>in</strong>t measurable sets. It is an elementary exercise <strong>in</strong> measure<br />

<strong>the</strong>ory to verify that a(W) -- b(W) = c(W). Now if c(W) > 0, take a sequence {Xn}n~=l<br />

<strong>in</strong> W and a sequence {An}n~=l <strong>of</strong> disjo<strong>in</strong>t measurable sets so that 0 < [IXn 1an II1 ~ c(W).<br />

Clearly c({xn 1Xn}) = 0, so {Xn 1X, } has weakly compact closure. The sequence {Xn 1 An }nCC_--I<br />

is equivalent to <strong>the</strong> unit vector basis <strong>of</strong> e 1 s<strong>in</strong>ce <strong>the</strong> An's are disjo<strong>in</strong>t and IlXn 1An II is<br />

bounded away from zero. This implies that a subsequence <strong>of</strong> {Xn}nCC=l is also equivalent<br />

to <strong>the</strong> unit vector basis <strong>of</strong> el. (Remark: If {Yn}n~ is bounded <strong>in</strong> any L1 (IX) space<br />

and <strong>the</strong>re is a sequence {Bn}n~__l <strong>of</strong> disjo<strong>in</strong>t measurable sets and b > 0 so that for all n,<br />

Y-~k#n llynlsk 111 ~< b/2 < b

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 19<br />

as T and <strong>the</strong> range <strong>of</strong> T is separable because X~ Y is separable. Thus it is enough to check<br />

that if Z is a separable space which conta<strong>in</strong>s co, <strong>the</strong>n <strong>the</strong>re is a projection <strong>of</strong> norm at most<br />

two from Z onto co. To see this, first let zn * be an extension <strong>of</strong> <strong>the</strong> nth unit vector <strong>in</strong> ~ 1 = Co *<br />

to a norm one element <strong>of</strong> Z*. Let d(., .) be a translation <strong>in</strong>variant metric on Z* which<br />

<strong>in</strong>duces <strong>the</strong> weak* topology on <strong>the</strong> unit ball <strong>of</strong> Z*; for example, if {Xn}n~ is dense <strong>in</strong> <strong>the</strong><br />

unit sphere <strong>of</strong> Z <strong>the</strong> metric can be def<strong>in</strong>ed by d(x*, y*) - y~n~__l 2-n I (x * - y*)(Xn)l. S<strong>in</strong>ce<br />

every weak* limit po<strong>in</strong>t <strong>of</strong> {Zn*}n~ belongs to <strong>the</strong> unit ball B <strong>of</strong> <strong>the</strong> annihilator <strong>of</strong> co <strong>in</strong> Z*,<br />

:r<br />

it follows that d (Zn*, B) --+ 0. Thus we can choose w n * <strong>in</strong> B so that d (z*, w n) --+ O, which<br />

means that Zn * -- W n * --+ 0 weak*. The formula Pz "-- {(z* - w,)(z)* }n=l ~ def<strong>in</strong>es a projection<br />

<strong>of</strong> norm at most two from Z onto co.<br />

That <strong>the</strong> space co is not separably )~-<strong>in</strong>jective for any )~ < 2 can be seen by consider<strong>in</strong>g<br />

P 1 for any projection P from c onto co. More <strong>in</strong>terest<strong>in</strong>g is that <strong>the</strong> separability assumption<br />

is needed: There is no projection from s onto co. Indeed, o<strong>the</strong>rwise we would have<br />

s ~ co (9 g~cc/co. But s is not isomorphic to a subspace <strong>of</strong> s This can be seen by<br />

prov<strong>in</strong>g that c0(R) embeds <strong>in</strong>to got/co but not <strong>in</strong>to s c0(R) admits no countable separat<strong>in</strong>g<br />

family <strong>of</strong> l<strong>in</strong>ear functionals, so it does not embed <strong>in</strong>to s To see that c0(R) embeds<br />

<strong>in</strong>to s take a family {Ar}rER <strong>of</strong> <strong>in</strong>f<strong>in</strong>ite subsets <strong>of</strong> N so that <strong>the</strong> <strong>in</strong>tersection <strong>of</strong> any<br />

two is f<strong>in</strong>ite (replace N by <strong>the</strong> rationals and for each r E IR consider a sequence <strong>of</strong> rationals<br />

which converges to r). Let Xr be <strong>the</strong> image <strong>in</strong> <strong>the</strong> quotient space s <strong>of</strong> <strong>the</strong> <strong>in</strong>dicator<br />

function <strong>of</strong> Ar. It is easy to check that {Xr}rEIR is 1-equivalent to <strong>the</strong> unit vector basis <strong>of</strong><br />

c0(R).<br />

The space co is <strong>the</strong> only (up to isomorphism) separable space which is separably <strong>in</strong>jective<br />

[43]. General C (K) spaces do have ano<strong>the</strong>r useful <strong>in</strong>to extension property" A compact<br />

operator T from a subspace Y <strong>of</strong> a <strong>Banach</strong> space X <strong>in</strong>to C(K) has an extension T to<br />

a compact operator from X <strong>in</strong>to C(K). Indeed, s<strong>in</strong>ce C(K) spaces have <strong>the</strong> BAR T can<br />

be approximated <strong>in</strong> <strong>the</strong> operator norm by operators <strong>of</strong> f<strong>in</strong>ite rank and hence we can write<br />

T = ~--~=0 oc Tn with each Tn <strong>of</strong> f<strong>in</strong>ite rank and II Zn II < 2 -n for n ~> 1. Therefore it is enough<br />

to observe that if S'X --+ C(K) has f<strong>in</strong>ite rank, <strong>the</strong>n S extends to a f<strong>in</strong>ite rank operator<br />

S" X --+ C(K) with IISII ~< (1 + e)llSII (where e > 0 is arbitrary). But by us<strong>in</strong>g partitions<br />

<strong>of</strong> unity and <strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong> small perturbations one checks that <strong>the</strong>re is a subspace E <strong>of</strong><br />

C (K) so that S X C E and d (E, g n) < 1 + ~, where n -- dim E < ec. Then E is (1 + e)-<br />

<strong>in</strong>jective and hence <strong>the</strong> desired extension <strong>of</strong> S exists. The same argument works when<br />

C(K) is replaced by any/2oc space (def<strong>in</strong>ed <strong>in</strong> Section 9).<br />

Every Loc(#) is isometric to C(K) for some compact Hausdorff space K. This follows<br />

from Gelfand <strong>the</strong>ory, s<strong>in</strong>ce Loc(#) is a commutative B* algebra with unit (see [19,<br />

Chapter 11]). Alternatively, it follows from lattice characterizations <strong>of</strong> C(K) spaces (see<br />

Section 5).<br />

The C (K) spaces play a special r61e <strong>in</strong> <strong>Banach</strong> space <strong>the</strong>ory because <strong>the</strong>y are a universal<br />

class: Every <strong>Banach</strong> space X is isometric to a subspace <strong>of</strong> some C(K) space. This can be<br />

seen by embedd<strong>in</strong>g X <strong>in</strong>to goc (F) for F appropriately large and apply<strong>in</strong>g <strong>the</strong> comment <strong>in</strong><br />

<strong>the</strong> previous paragraph. Alternatively, X isometrically embeds via evaluation <strong>in</strong>to C(K)<br />

with K <strong>the</strong> unit ball <strong>of</strong> X* with <strong>the</strong> weak* topology. This approach is preferable because<br />

<strong>the</strong> unit ball <strong>of</strong> X* is weak* metrizable when X is separable. In fact, every separable<br />

<strong>Banach</strong> space X embeds isometrically <strong>in</strong>to C[0, 1]. First, C(A) embeds isometrically <strong>in</strong>to<br />

C[0, 1 ], where A is <strong>the</strong> usual "middle thirds" Cantor set <strong>in</strong> [0, 1 ], by extend<strong>in</strong>g a cont<strong>in</strong>uous

20 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

function on A aff<strong>in</strong>ely on <strong>the</strong> component <strong>in</strong>tervals <strong>of</strong> <strong>the</strong> complement <strong>of</strong> A <strong>in</strong> [0, 1 ]. Next,<br />

us<strong>in</strong>g that A is homeomorphic to {0, 1 }r~, it can be shown that every compact metric space<br />

K is <strong>the</strong> cont<strong>in</strong>uous image <strong>of</strong> A. Build a tree structure K(il, ie ..... <strong>in</strong>): {il, ie ..... <strong>in</strong>}<br />

{0, 1}n; n = 1, 2 .... } <strong>of</strong> nonempty closed subsets <strong>of</strong> K so that each K(il, i2 ..... <strong>in</strong>) is<br />

<strong>the</strong> union <strong>of</strong> K (il, ie ..... <strong>in</strong>, 0) and K (il, i2 ..... <strong>in</strong>, 1), K -- K0 U K1, and <strong>the</strong> maximum<br />

diameter <strong>of</strong> K(il, i2 .... , <strong>in</strong>) as {il, ie ..... <strong>in</strong>} varies over {0, 1} n tends to zero as n --+ cx~.<br />

This is not hard to do us<strong>in</strong>g compactness <strong>of</strong> K. The map def<strong>in</strong>ed by {ij}7-1 ~ K(il) N<br />

K(il, i2) N K(il, ie, i3) N... is <strong>the</strong>n a cont<strong>in</strong>uous surjection from A = {0, 1} N onto K.<br />

Hav<strong>in</strong>g gotten a cont<strong>in</strong>uous mapp<strong>in</strong>g g from A onto K, one embeds C(K) <strong>in</strong>to C(A) by<br />

<strong>the</strong> formula T x = x o g, x E C (K).<br />

The isometric classification <strong>of</strong> C(K) spaces is easy and can be found <strong>in</strong> beg<strong>in</strong>n<strong>in</strong>g texts<br />

(such as [12, Theorem 231]): The space C(K) is isometric to C(H) if and only if K is<br />

homeomorphic to H. This follows from <strong>the</strong> identification <strong>of</strong> <strong>the</strong> extreme po<strong>in</strong>ts <strong>of</strong> <strong>the</strong> unit<br />

ball <strong>of</strong> C(K)* as <strong>the</strong> evaluation functionals at po<strong>in</strong>ts <strong>of</strong> K (as well as multiples <strong>of</strong> <strong>the</strong>se<br />

functionals by scalars <strong>of</strong> magnitude one). The much harder problem <strong>of</strong> <strong>the</strong> isomorphic classification<br />

<strong>of</strong> C (K) spaces has been accomplished <strong>in</strong> <strong>the</strong> separable case (that is, for compact<br />

metric spaces): If K is an uncountable compact metric space <strong>the</strong>n C(K) is isomorphic to<br />

C (0, 1). If K is a countable compact metric space <strong>the</strong>n K is homeomorphic to <strong>the</strong> space<br />

[ 1, or] <strong>of</strong> all ord<strong>in</strong>als up to <strong>the</strong> ord<strong>in</strong>al ot for some countable ord<strong>in</strong>al c~ <strong>in</strong> <strong>the</strong> order topology.<br />

C (1, or) is isomorphic to C (1, t) when ot < fl if and only if fl < c( ~ See [40] for fur<strong>the</strong>r<br />

discussion <strong>of</strong> <strong>the</strong> separable case. It seems a hopeless task to get an isomorphic classification<br />

<strong>of</strong> general C(K) spaces, but some <strong>in</strong>formation is conta<strong>in</strong>ed <strong>in</strong> [44].<br />

We have already mentioned <strong>in</strong>explicitly that <strong>the</strong> dual <strong>of</strong> an L l(#) space is isometric to<br />

a C (K) space. Similarly, <strong>the</strong> dual <strong>of</strong> a C (K) is isometric to L 1 (//~) for some measure/x.<br />

The usual representation (see [18]) <strong>of</strong> <strong>the</strong> dual <strong>of</strong> C(K) is <strong>the</strong> space M(K) <strong>of</strong> f<strong>in</strong>ite signed<br />

measures on <strong>the</strong> sigma algebra <strong>of</strong> Baire subsets <strong>of</strong> K (that is, <strong>the</strong> sigma algebra generated<br />

by <strong>the</strong> closed G~ subsets <strong>of</strong> K). It is a rout<strong>in</strong>e exercise us<strong>in</strong>g <strong>the</strong> Radon-Nikod3)m <strong>the</strong>o-<br />

rem to verify that if {/x• }•<br />

is a maximal family <strong>of</strong> mutually s<strong>in</strong>gular Baire probability<br />

measures on K, <strong>the</strong>n M(K) is isometric to (Y-~yer L1 (/zy))l.<br />

Once <strong>the</strong> duals <strong>of</strong> C(K) and <strong>of</strong> LI(/Z) are classified, it is natural to ask what are <strong>the</strong><br />

preduals <strong>of</strong> C(K) and L1 (/z)? It turns out that every isometric predual <strong>of</strong> a space C(K)<br />

is isometric to L1 (/z) for some measure # and that all preduals <strong>of</strong> C (K) are mutually isometric.<br />

On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> isomorphic preduals <strong>of</strong>, for example, g~, form a rich class<br />

<strong>of</strong> spaces which are still not well understood. Preduals <strong>of</strong> L1 (/z) spaces are even less well<br />

understood. Here we mention only that <strong>the</strong> space ~1 has uncountably many mutually nonisomorphic<br />

preduals among <strong>the</strong> C(K) spaces (C(K)* is isometric to el if K is a countable<br />

compact space).<br />

5. <strong>Banach</strong> lattices<br />

Most <strong>of</strong> <strong>the</strong> <strong>Banach</strong> spaces that appear naturally <strong>in</strong> analysis carry structure <strong>in</strong> addition<br />

to <strong>the</strong>ir structure as <strong>Banach</strong> spaces. It tums out that this additional structure can affect<br />

properties which are def<strong>in</strong>ed purely <strong>in</strong> terms <strong>of</strong> <strong>Banach</strong> space concepts (such as l<strong>in</strong>ear<br />

operators and duality). An extra structure that classical spaces such as Lp and C (K) spaces

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 21<br />

possess is that <strong>of</strong> a <strong>Banach</strong> lattice, which is a <strong>Banach</strong> space over <strong>the</strong> reals that is equipped<br />

with a partial order ~< for which x v y and x A y exist for all vectors x, y, and such that <strong>the</strong><br />

positive cone is closed under addition and multiplication by nonnegative real numbers and<br />

<strong>the</strong> order is connected to <strong>the</strong> norm by <strong>the</strong> condition that Ixl ~< lYl =r Ilxll ~< IlYlI, where <strong>the</strong><br />

absolute value is def<strong>in</strong>ed by Ix l = x v (-x).<br />

A l<strong>in</strong>ear mapp<strong>in</strong>g from a <strong>Banach</strong> lattice to a <strong>Banach</strong> lattice is positive if it carries positive<br />

vectors to positive vectors. This is equivalent to say<strong>in</strong>g that <strong>the</strong> mapp<strong>in</strong>g is order preserv<strong>in</strong>g.<br />

It is easy to see that a positive l<strong>in</strong>ear mapp<strong>in</strong>g is cont<strong>in</strong>uous, so we call <strong>the</strong>m positive<br />

operators. If a positive operator preserves <strong>the</strong> lattice operations, it is called a lattice homomorphism.<br />

It is clear that a one-to-one surjective positive operator T whose <strong>in</strong>verse is<br />

positive is a lattice isomorphism; that is, both T and T-1 are lattice homomorphisms. The<br />

dual <strong>of</strong> a <strong>Banach</strong> lattice X is aga<strong>in</strong> a <strong>Banach</strong> lattice under <strong>the</strong> standard order<strong>in</strong>g on X* <strong>in</strong><br />

which <strong>the</strong> positive l<strong>in</strong>ear functionals form <strong>the</strong> positive cone. With this def<strong>in</strong>ition it is easy<br />

to see that <strong>the</strong> canonical mapp<strong>in</strong>g from X <strong>in</strong>to X** is positive [ 15, 1.a.2].<br />

While <strong>the</strong>re are <strong>in</strong>terest<strong>in</strong>g functional analytical topics (such as positive operators and<br />

lattice homomorphisms) which are special to <strong>Banach</strong> lattices, we are mostly <strong>in</strong>terested<br />

here <strong>in</strong> <strong>the</strong> <strong>Banach</strong> space properties <strong>of</strong> <strong>Banach</strong> lattices and concentrate on those lattice<br />

properties which affect <strong>the</strong> l<strong>in</strong>ear topological or geometry <strong>of</strong> <strong>the</strong> underly<strong>in</strong>g <strong>Banach</strong> space.<br />

The basic reference for this aspect <strong>of</strong> <strong>Banach</strong> lattices is [ 15]. For a general <strong>in</strong>troduction to<br />

<strong>Banach</strong> lattices which covers <strong>the</strong> basic <strong>the</strong>ory as well as some fairly recent material see [1 ].<br />

The simplest examples <strong>of</strong> <strong>Banach</strong> lattices are <strong>the</strong> spaces with a monotonely unconditional<br />

basis under <strong>the</strong> po<strong>in</strong>twise order on <strong>the</strong> coefficients, which as we have seen are better<br />

behaved, or at least more regularly behaved, than general <strong>Banach</strong> spaces. It turns out that<br />

much <strong>of</strong> <strong>the</strong> structure <strong>the</strong>ory for <strong>the</strong>se spaces carries over to <strong>Banach</strong> lattices.<br />

It was already mentioned that <strong>the</strong> Lp and C(K) spaces are <strong>Banach</strong> lattices. O<strong>the</strong>r examples<br />

are <strong>the</strong> Orlicz spaces and <strong>the</strong> Lorentz spaces. An Orliczfunction is an even convex<br />

function on R which is zero at zero and tends to <strong>in</strong>f<strong>in</strong>ity at <strong>in</strong>f<strong>in</strong>ity. If # is a measure and<br />

M is an Orlicz function, <strong>the</strong> space LM(/Z) is <strong>the</strong> collection <strong>of</strong> all/z-measurable functions<br />

f for which <strong>the</strong>re exists C > 0 so that f M(f/C)d/z < ~. Then II f IIM is def<strong>in</strong>ed to be<br />

<strong>the</strong> <strong>in</strong>fimum <strong>of</strong> those C > 0 for which f M(f/C) d/z < 1. This is a norm on LM(/z) which<br />

makes LM(/z) <strong>in</strong>to a <strong>Banach</strong> lattice. If/z is count<strong>in</strong>g measure on N, LM(/z) is called an<br />

Orlicz sequence space and is denoted by gM. If 1 ~< p < ~ and W is a positive non<strong>in</strong>creas<strong>in</strong>g<br />

cont<strong>in</strong>uous function on (0, ~) so that W(t) --+ ~ as t ~ 0+, W(t) --+ 0 as t --+ c~,<br />

fl W(t) dt -- 1, and fo W(t) dt -- cx~, <strong>the</strong> Lorentz space LW, p(/z) is <strong>the</strong> space <strong>of</strong> all/z<br />

measurable functions f for which IlflIw, p "-(f~xz f,(t)PW(t)dt)l/p ' where f* is <strong>the</strong><br />

decreas<strong>in</strong>g rearrangement <strong>of</strong> Ifl. The space Lw, p(/z) is a <strong>Banach</strong> lattice under <strong>the</strong> norm<br />

II f II W,p. Lorentz sequence spaces are def<strong>in</strong>ed <strong>in</strong> an analogous fashion.<br />

The Orlicz spaces L M (/z) and <strong>the</strong> Lorentz spaces L w, p (/z), like <strong>the</strong> spaces Lp (/z), are<br />

symmetric lattices which are ideals <strong>in</strong> <strong>the</strong> lattice <strong>of</strong> all/z-measurable functions. Here X is<br />

symmetric means that if x E X and y is a/z-measurable function for which x* = y*, <strong>the</strong>n y<br />

is <strong>in</strong> X and IlYllx = Ilxllx. A subspace Y <strong>of</strong> a lattice X is an ideal provided x 6 X, y E Y,<br />

and Ix l ~< l Yl =~ x E Y. Symmetric lattice ideals play an important role <strong>in</strong> <strong>in</strong>terpolation<br />

<strong>the</strong>ory; see Section 11 for a short <strong>in</strong>troduction. (Some sources call a symmetric lattice a<br />

rearrangement <strong>in</strong>variant space; o<strong>the</strong>r references use "rearrangement <strong>in</strong>variant space" to

22 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

mean "symmetric lattice ideal", sometimes with extra conditions. That is why we avoid<br />

<strong>the</strong> term "rearrangement <strong>in</strong>variant" <strong>in</strong> this article.)<br />

It is no accident that <strong>in</strong> <strong>the</strong> examples <strong>of</strong> <strong>Banach</strong> lattices given above <strong>the</strong> lattice order<strong>in</strong>g<br />

is just <strong>the</strong> natural po<strong>in</strong>twise a.e. order<strong>in</strong>g on a space <strong>of</strong> (equivalence classes <strong>of</strong>) real valued<br />

functions on a measure space (for C(K) <strong>the</strong> measure is count<strong>in</strong>g measure on K and for<br />

spaces with unconditional basis <strong>the</strong> measure is count<strong>in</strong>g measure on N). We call such a<br />

<strong>Banach</strong> lattice a <strong>Banach</strong> lattice <strong>of</strong> functions or a <strong>Banach</strong> lattice <strong>of</strong> #-measurable functions<br />

if it is important to specify <strong>the</strong> measure. There are representation <strong>the</strong>orems which say that<br />

<strong>the</strong>re is essentially no loss <strong>of</strong> generality <strong>in</strong> consider<strong>in</strong>g only <strong>Banach</strong> lattices <strong>of</strong> functions.<br />

The most classical <strong>of</strong> <strong>the</strong>se representation <strong>the</strong>orems, called <strong>the</strong> Kakutani representation<br />

<strong>the</strong>orem, gives abstract <strong>Banach</strong> lattice characterizations <strong>of</strong> L1 (/z) and C (K) spaces. A <strong>Banach</strong><br />

lattice X is an abstract Lp space, 1 ~< p < cx~, provided IIx + yll p = Ilxll p + Ilyll p<br />

whenever x and y are disjo<strong>in</strong>t; that is, Ixl A lyl - 0. The Lp version <strong>of</strong> <strong>the</strong> Kakutani representation<br />

<strong>the</strong>orem says that an abstract Lp space is lattice isometric to Lp(#) for some<br />

measure lz (see [ 15, 1.b.2] or [ 1, Theorem 12.26]). Moreover, <strong>the</strong> measure/x can be chosen<br />

to be a f<strong>in</strong>ite measure if <strong>the</strong> abstract L p space X has a weak order unit; that is, a vector<br />

u ) 0 so that u A Ix l- 0 only when x is <strong>the</strong> zero vector. A <strong>Banach</strong> lattice X is an abstract<br />

M space provided IIx + Yll - Ilxll v IlYll whenever x and y are disjo<strong>in</strong>t. The C(K)<br />

representation <strong>the</strong>orem says that an abstract M space is lattice isometric to a sublattice<br />

<strong>of</strong> C(K) for some compact Hausdorff space K (see [15, 1.b.6]). Moreover, if <strong>the</strong> abstract<br />

M space has a strong order unit (that is, a vector u ~> 0 such that <strong>the</strong> unit ball <strong>of</strong> X is <strong>the</strong><br />

order <strong>in</strong>terval [-u, u] := {x: -u ~< x ~< u}), <strong>the</strong>n M is lattice isometric to C(K) itself via<br />

an isomorphism which maps u to Ilu II 1K.<br />

Suppose that <strong>the</strong> <strong>Banach</strong> lattice X admits a strictly positive functional; that is, a positive<br />

l<strong>in</strong>ear functional u* such that u*(Ixl) > 0 for every nonzero vector x <strong>in</strong> X. Def<strong>in</strong>e an<br />

(<strong>in</strong>equivalent) norm I1" Ilu* on X by Ilxllu* := u*(Ixl). This is obviously an (<strong>in</strong>complete)<br />

abstract L1 norm on X from which it follows that <strong>the</strong> completion Xu, <strong>of</strong> (X, IIx Ilu*) is an<br />

abstract L 1 space. S<strong>in</strong>ce <strong>the</strong> formal <strong>in</strong>clusion operator from X to Xu, is an <strong>in</strong>jective lattice<br />

homomorphism, <strong>the</strong> Kakutani representation <strong>the</strong>orem yields that X can be thought <strong>of</strong> as<br />

a <strong>Banach</strong> lattice <strong>of</strong> functions. If X is separable, so is Xu,, and it follows from comments<br />

made <strong>in</strong> Section 4 that Xu, is isometric and order equivalent to a sublattice <strong>of</strong> L l (0, 1), so<br />

that X can be represented as a <strong>Banach</strong> lattice <strong>of</strong> Lebesgue measurable functions on <strong>the</strong> unit<br />

<strong>in</strong>terval.<br />

Not every <strong>Banach</strong> lattice admits a strictly positive functional (co(F) with F uncountable<br />

is a counterexample), but every separable <strong>Banach</strong> lattice X does. Indeed, take {Xn}n~=l<br />

dense <strong>in</strong> <strong>the</strong> unit sphere <strong>of</strong> X and for each n pick a norm one functional x n * which achieves<br />

its norm at Xn. It is easy to check that u* "- ~ 2 -n Ix~*l is a strictly positive functional.<br />

From this comment and <strong>the</strong> representation above it follows that any lattice <strong>in</strong>equality <strong>in</strong>volv<strong>in</strong>g<br />

f<strong>in</strong>itely many vectors which is true for lattices <strong>of</strong> functions must be true <strong>in</strong> a general<br />

<strong>Banach</strong> lattice. This elim<strong>in</strong>ates <strong>the</strong> tedium <strong>of</strong> verify<strong>in</strong>g "obvious" <strong>in</strong>equalities (for example,<br />

V( ~i=1 " ~:iXi 9 ~i -- -+'1} -~- ff-~i=l ]Xi[) directly from <strong>the</strong> axioms for a <strong>Banach</strong> lattice.<br />

The representation <strong>of</strong> <strong>Banach</strong> lattices as lattices <strong>of</strong> functions suggests that abstract L1<br />

spaces are particularly important lattices. The abstract M spaces also arise naturally <strong>in</strong> <strong>the</strong><br />

general <strong>the</strong>ory. If u is a positive vector <strong>in</strong> a <strong>Banach</strong> lattice X and Xu is <strong>the</strong> l<strong>in</strong>ear span <strong>of</strong><br />

<strong>the</strong> order <strong>in</strong>terval [-u, u] with [-u, u] taken as <strong>the</strong> unit ball, <strong>the</strong>n Xu is easily seen to be

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 23<br />

an abstract M space with strong order unit u (Xu is complete because [-u, u] is closed <strong>in</strong><br />

<strong>the</strong> <strong>Banach</strong> space X).<br />

A <strong>Banach</strong> lattice is order complete or Dedek<strong>in</strong>d complete if every nonempty subset<br />

which is bounded above has a least upper bound. A dual <strong>Banach</strong> lattice is order complete;<br />

<strong>in</strong> fact, any norm bounded upwarded directed net <strong>in</strong> a dual <strong>Banach</strong> lattice converges<br />

weak* to <strong>the</strong> least upper bound <strong>of</strong> <strong>the</strong> net.<br />

Suppose that X is order complete and u ~> 0 with llu I] = 1. We look a bit more closely<br />

at <strong>the</strong> C(K) space which is lattice isometric to <strong>the</strong> abstract M space Xu. Given x ~> 0,<br />

def<strong>in</strong>e <strong>the</strong> support <strong>of</strong> x by S(x) := supn(nx)/x u. The supremum exists because X is<br />

order complete. The support S(x) is a component <strong>of</strong> u. (A vector 0 ~< y ~< u is called a<br />

component <strong>of</strong> u [or simply a component if u is understood] provided y is disjo<strong>in</strong>t from<br />

u - y.) If T:X, --+ C(K) is a lattice isometry with Tu = 1K, <strong>the</strong>n Tx is an <strong>in</strong>dicator<br />

function if and only if x is a component. Of course, if A C K, 1 a is <strong>in</strong> C (K) if and only if<br />

A is clopen (i.e., both open and closed). So <strong>the</strong> components form a Boolean algebra (a fact<br />

which is also easy to verify directly) which is complete because X is order complete. Thus<br />

<strong>the</strong> clopen subsets <strong>of</strong> K are also a complete Boolean algebra. An important fact is that <strong>the</strong><br />

clopen subsets <strong>of</strong> K form a base for <strong>the</strong> topology <strong>of</strong> K. One way to see this is to observe<br />

that if 0 ~< x with x <strong>in</strong> Xu and t > 0 satisfies y := (x - tu) v 0 ~ 0, <strong>the</strong>n 0 ~< tS(y) 0] conta<strong>in</strong>s a nonempty clopen subset. S<strong>in</strong>ce K is compact this implies that <strong>the</strong> clopen<br />

subsets <strong>of</strong> K form a base for <strong>the</strong> topology. Us<strong>in</strong>g this and <strong>the</strong> completeness <strong>of</strong> <strong>the</strong> Boolean<br />

algebra <strong>of</strong> clopen sets it is a simple exercise to prove that C (K) is itself an order complete<br />

<strong>Banach</strong> lattice. As was po<strong>in</strong>ted out <strong>in</strong> Section 4, this implies that C (K) is 1-<strong>in</strong>jective.<br />

From <strong>the</strong> discussion <strong>in</strong> <strong>the</strong> previous paragraph we can deduce: IfE is af<strong>in</strong>ite dimensional<br />

subspace <strong>of</strong> an order complete lattice X and ~ > O, <strong>the</strong>n <strong>the</strong>re is a f<strong>in</strong>ite dimensional sublattice<br />

F <strong>of</strong> X and an automorphism T <strong>of</strong> X so that E C T F and ]l I - T ]l < e. Given <strong>the</strong><br />

subspace E, take positive vectors x l .... , xn <strong>in</strong> X whose span conta<strong>in</strong>s E and normalized<br />

so that u := max/xi has norm one. Then F C Xu and Xu is isometric to a C(K) space for<br />

which <strong>the</strong> clopen subsets <strong>of</strong> K form a base for <strong>the</strong> topology, which means that <strong>the</strong> span <strong>of</strong><br />

<strong>in</strong>dicator functions <strong>of</strong> clopen sets is dense <strong>in</strong> C(K). Thus fix<strong>in</strong>g a basis yl ..... yk for E<br />

and 3 > 0, we get a subspace F <strong>of</strong> Xu spanned by disjo<strong>in</strong>t vectors and a vector u <strong>in</strong> F with<br />

]lull = 1 so that for each 1 ~< i ~< k, <strong>the</strong>re is a vector Xi SO that ]Xi -- Zi] ~ (~U (and hence<br />

Ilxi - zi [I ~< ~). Now apply <strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong> small perturbations.<br />

A <strong>Banach</strong> lattice is order cont<strong>in</strong>uous if every downward directed net whose greatest<br />

lower bound is zero converges <strong>in</strong> norm (or weakly; it is <strong>the</strong> same) to zero. This is equivalent<br />

to say<strong>in</strong>g that every order bounded <strong>in</strong>creas<strong>in</strong>g sequence converges <strong>in</strong> norm (necessarily<br />

to <strong>the</strong> least upper bound <strong>of</strong> <strong>the</strong> sequence) (see [15, 1.a.8] or [1, Theorem 12.9] or <strong>the</strong><br />

beg<strong>in</strong>n<strong>in</strong>g <strong>of</strong> <strong>the</strong> argument below). It is also easy to check that an order cont<strong>in</strong>uous <strong>Banach</strong><br />

lattice is order complete [15, 1.a.8]. A <strong>Banach</strong> lattice is not order cont<strong>in</strong>uous if and only<br />

if it conta<strong>in</strong>s a sequence <strong>of</strong> disjo<strong>in</strong>t positive vectors which is equivalent to <strong>the</strong> unit vector<br />

basis for co and is bounded above. The "if" direction is clear. If X is not order cont<strong>in</strong>uous,<br />

one gets an upward directed net <strong>of</strong> positive vectors which is bounded above by, say, x,<br />

with ]ix ]l = 1. The net cannot converge <strong>in</strong> norm, so one gets 0 ~< x l ~< x2 ~< ... ~< x so that

24 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

a := <strong>in</strong>fn Ilxn+~ - Xn II > 0. Let Yn := Xn+l -- Xn ~ O. So we have for every n:<br />

Yn >~ O, IlYn II ~ a, ~ Yk ~< x. (2)<br />

k--1<br />

(Although we do not need it here because we want to "disjo<strong>in</strong>tify" <strong>the</strong> yn'S, it is worth<br />

notic<strong>in</strong>g that (2) implies that Yn --+ 0 weakly and hence {Yn}n~__l has a basic subsequence,<br />

and that any basic subsequence <strong>of</strong> {yn}n~__l is equivalent to <strong>the</strong> unit vector basis for co.)<br />

Take Yn* <strong>in</strong> <strong>the</strong> unit sphere <strong>of</strong> X* so that Y*(Yn)= [[Yn[[. By replac<strong>in</strong>g y~* with [Yn*[ if<br />

necessary, we may assume that y~* ~> 0. For each n <strong>the</strong> nonnegative sum Y~m Y,~ (Ym) is at<br />

most I[x [I. It is an elementary exercise <strong>in</strong> comb<strong>in</strong>atorial reason<strong>in</strong>g to deduce from this that<br />

for any E > 0 <strong>the</strong>re is a subsequence {zk}~__l "-- {Ynk }~--1 <strong>of</strong> {yn}n~__l so that for each k > j,<br />

Z k*(Zj) < 2-J6 2, where z k * "-- Ynk" * Thus for each n,<br />

(Zn+l--6-1~Zk)k=l V0 Zn+ 1 Zn+ l -- 6 1 Zk >~ a - e.<br />

k=l<br />

This means that <strong>the</strong> zn's have big disjo<strong>in</strong>t pieces. More precisely, let e = a/4 and set<br />

(Z +l and Wn := (Vn -- EX) V O.<br />

Then <strong>the</strong> Wn'S are pairwise disjo<strong>in</strong>t positive vectors smaller than x with norms bounded<br />

away from zero.<br />

It is easy to see that {Wn}n~__l must be equivalent to <strong>the</strong> unit vector basis <strong>of</strong> co. This<br />

completes <strong>the</strong> pro<strong>of</strong>, but note that if X is order complete <strong>the</strong> mapp<strong>in</strong>g en ~ Wn from co<br />

<strong>in</strong>to X extends to a mapp<strong>in</strong>g from <strong>the</strong> positive cone <strong>of</strong> s to X by def<strong>in</strong><strong>in</strong>g {~n6n }n~__l<br />

sup n an Wn; this mapp<strong>in</strong>g extends to an order isomorphism from s <strong>in</strong>to X.<br />

O<strong>the</strong>r useful characterizations <strong>of</strong> order cont<strong>in</strong>uous <strong>Banach</strong> lattices are given by <strong>the</strong> follow<strong>in</strong>g<br />

(see [15, 1.b.16] or [1, Theorem 12.9]): X is order cont<strong>in</strong>uous if and only if every<br />

order <strong>in</strong>terval is weakly compact if and only if X is an ideal <strong>in</strong> X**.<br />

S<strong>in</strong>ce c is not order cont<strong>in</strong>uous but is isomorphic to <strong>the</strong> order cont<strong>in</strong>uous <strong>Banach</strong> lattice<br />

co, <strong>the</strong>re is not a l<strong>in</strong>ear topological characterization <strong>of</strong> order cont<strong>in</strong>uity. While it is only a<br />

sufficient condition for order cont<strong>in</strong>uity <strong>of</strong> X that co not embed isomorphically <strong>in</strong>to X, this<br />

condition is only a bit too strong as arguments similar to those given above show that if co<br />

embeds <strong>in</strong>to X <strong>the</strong>n it embeds as a sublattice (see [1, Theorem 14.12]).<br />

The functional representation <strong>the</strong>orems for order cont<strong>in</strong>uous <strong>Banach</strong> lattices are stronger<br />

and more useful than <strong>the</strong> representation already mentioned for <strong>Banach</strong> lattices which have<br />

a strictly positive functional. Here we just <strong>in</strong>dicate what is go<strong>in</strong>g on and refer to [15] for<br />

details. However, <strong>the</strong> motivated reader is encouraged to work out <strong>the</strong> details for himself or<br />

herself (partly because <strong>the</strong> discussion <strong>in</strong> [15] wanders unnecessarily). First, it is not hard to<br />

show that an order cont<strong>in</strong>uous <strong>Banach</strong> lattice which has a weak order unit u also admits a<br />

strictly positive functional u* (see [15, 1.b.15] or [1, 12.14]) and thus can be thought <strong>of</strong> as

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 25<br />

a space <strong>of</strong> <strong>in</strong>tegrable functions on a f<strong>in</strong>ite measure space (S-2,#) with u* (x) = f x d# for<br />

x <strong>in</strong> X. One can assume that u* (u) = 1 = Ilu II and Ilu* II ~< 2. S<strong>in</strong>ce X is order cont<strong>in</strong>uous,<br />

one has for each x ~> 0 <strong>in</strong> X that (nu)/x x converges to x as n --+ cx~, so that Xu is dense<br />

<strong>in</strong> X. By replac<strong>in</strong>g <strong>the</strong> underly<strong>in</strong>g a-algebra with <strong>the</strong> smallest cr-algebra for which all<br />

<strong>the</strong> functions <strong>in</strong> X are measurable, it can be assumed that X is dense <strong>in</strong> L1 (#). Then<br />

necessarily u > 0 a.e. By replac<strong>in</strong>g <strong>the</strong> measure # with u d# and functions f <strong>in</strong> X by f/u,<br />

one can assume u -= 1. Now it is possible to recover <strong>the</strong>/z-measurable sets and <strong>the</strong> measure<br />

#. From <strong>the</strong> density <strong>of</strong> X <strong>in</strong> L 1 (/Z) it follows that <strong>the</strong> <strong>in</strong>dicator function <strong>of</strong> a measurable set<br />

is an element x <strong>in</strong> Xu which is a component <strong>of</strong> u. From this it is essentially obvious that <strong>the</strong><br />

L~(#) norm on Xu agrees with its abstract M space norm. If we represent Xu as a C(K)<br />

space, <strong>the</strong>n <strong>the</strong> sets <strong>of</strong> positive/z measure are mapped onto <strong>the</strong> nonempty clopen subsets <strong>of</strong><br />

K <strong>in</strong> an obvious way so that we can represent X and L l(#) as function spaces on K and<br />

transfer <strong>the</strong> measure # to K. We <strong>the</strong>n get that C(K) C X C L1 (#) with both <strong>in</strong>clusions<br />

hav<strong>in</strong>g dense range. Moreover, C (K) = L~ (#) and every/z-measurable set is equal/z-a.e.<br />

to a clopen subset <strong>of</strong> K. F<strong>in</strong>ally, X is an ideal <strong>in</strong> L1 (/z) (use aga<strong>in</strong> order cont<strong>in</strong>uity and <strong>the</strong><br />

fact that X conta<strong>in</strong>s all <strong>in</strong>dicator functions).<br />

One consequence <strong>of</strong> this representation is: Let X be an order cont<strong>in</strong>uous <strong>Banach</strong> lattice<br />

which has a weak order unit. A closed subspace Y <strong>of</strong> X ei<strong>the</strong>r embeds <strong>in</strong>to L 1 (/z) for some<br />

measure/z or conta<strong>in</strong>s a normalized basic sequence which is equivalent to (even equal to<br />

a small perturbation <strong>of</strong>) a disjo<strong>in</strong>t sequence. For a pro<strong>of</strong> see [ 15, 1.c.8] or modify <strong>the</strong> pro<strong>of</strong><br />

<strong>of</strong> <strong>the</strong> dichotomy pr<strong>in</strong>ciple discussed <strong>in</strong> Section 4. By apply<strong>in</strong>g L1 <strong>the</strong>ory discussed <strong>in</strong><br />

Section 4 one gets that a subspace <strong>of</strong> an order cont<strong>in</strong>uous <strong>Banach</strong> lattice is reflexive if and<br />

only if it does not conta<strong>in</strong> a subspace isomorphic to ei<strong>the</strong>r s or co and that a nonreflexive<br />

<strong>Banach</strong> lattice has a sublattice which is order isomorphic to g l or co. The representation<br />

can also be used to prove that a <strong>Banach</strong> lattice which does not conta<strong>in</strong> a copy <strong>of</strong> co must<br />

be weakly sequentially complete [15, 1.c.4].<br />

Although we have used here <strong>the</strong> representation <strong>the</strong>orem for abstract L 1 spaces, it should<br />

be mentioned that s<strong>in</strong>ce an abstract L1 space is necessarily order cont<strong>in</strong>uous (every normalized<br />

disjo<strong>in</strong>t sequence is obviously 1-equivalent to <strong>the</strong> unit vector basis <strong>of</strong> ~ 1), some <strong>of</strong><br />

<strong>the</strong> ideas (particularly build<strong>in</strong>g components <strong>in</strong> X) can be used to prove Kakutani's representation<br />

<strong>of</strong> an abstract L 1 space which has a weak order unit as a space L 1 (/z) for some<br />

probability measure/z.<br />

Expressions such as (EnL1 ]Xn]P) 1/p, 1

26 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

By work<strong>in</strong>g <strong>in</strong> <strong>the</strong> abstract M space Xu with u "-- ~--~nL 1 Ixnl, <strong>the</strong> C(K) space case gives <strong>the</strong><br />

general case. We can <strong>the</strong>n <strong>in</strong>terpret scalar <strong>in</strong>equalities <strong>in</strong>volv<strong>in</strong>g such expressions <strong>in</strong> general<br />

lattices (ra<strong>the</strong>r than just <strong>in</strong> lattices <strong>of</strong> functions). For example, Kh<strong>in</strong>tch<strong>in</strong>e's <strong>in</strong>equality,<br />

discussed <strong>in</strong> Section 4, reads<br />

A p ~ IXn I 2 ~< E 6nXn<br />

n=l<br />

n=l<br />

Bp [Xn[ 2<br />

n=l<br />

1/2<br />

(3)<br />

Auxiliary lattices such as X (~p) can now be def<strong>in</strong>ed and <strong>the</strong> "obvious" dualities checked.<br />

X (g2) is especially noteworthy, s<strong>in</strong>ce it can be regarded as <strong>the</strong> complexification <strong>of</strong> X by<br />

regard<strong>in</strong>g ~2 as C and def<strong>in</strong><strong>in</strong>g multiplication by complex scalars <strong>in</strong> <strong>the</strong> obvious way (see<br />

[15, p. 43] for details).<br />

Start<strong>in</strong>g from <strong>the</strong> scalar identity Iotl~ 1-~ = <strong>in</strong>fc>oOCl/~ + (1 -O)C~/(~ one<br />

gets <strong>in</strong> a similar way an <strong>in</strong>terpretation for <strong>the</strong> expression Ixl~ ~-~ and that<br />

IIIxl~176 ~ Ilxll~ 1-~ (4)<br />

Also, if 1 ~< p < q ~< cx~, 1/r "--O/p + (1 -O)/q with 0 < 0 < 1 and Ctl ..... OtN are<br />

positive scalars, <strong>the</strong>n for any collection x l ..... xu <strong>of</strong> vectors <strong>in</strong> a <strong>Banach</strong> lattice we have<br />

that<br />

otnlxnl r

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 27<br />

The smallest such M is denoted by M(P)(T). Clearly M(1)(T) -- IITII. Similarly, if for a<br />

l<strong>in</strong>ear mapp<strong>in</strong>g T from a <strong>Banach</strong> lattice <strong>in</strong>to a <strong>Banach</strong> space <strong>the</strong> <strong>in</strong>equality<br />

(i IlTx. IIp<br />

n=l<br />

~M Ixnl p<br />

n--I<br />

1/p<br />

(8)<br />

always holds for some constant M, <strong>the</strong>n T is called p-concave and <strong>the</strong> smallest such M<br />

is denoted by M(p)(T). Clearly M(~)(T) = [[TI[. Evidently if T is p-convex [p-concave]<br />

<strong>the</strong>n it is bounded and M(P)(T) ~ [[TI[ [M(p)(T) ~ [[TI[]. It is not hard to check <strong>the</strong> identities<br />

M(P)(T *) = M(p,)(T) and M(P)(T) -- M(p,)(T*), where 1/p + 1/p* = 1. As functions<br />

<strong>of</strong> p, M(P)(T) is nondecreas<strong>in</strong>g and M(p)(T) is non<strong>in</strong>creas<strong>in</strong>g (see [15, 1.d.5] or<br />

write down a pro<strong>of</strong> for Lp(lZ ) and see that <strong>the</strong> H61der type <strong>in</strong>equalities (4), (5) allow a<br />

translation to <strong>the</strong> lattice sett<strong>in</strong>g).<br />

A <strong>Banach</strong> lattice X is called p-convex (p-concave) if <strong>the</strong> identity operator Ix on X is p-<br />

convex (p-concave) and we <strong>the</strong>n def<strong>in</strong>e M(P)(X) := M(P)(Ix) and M(p)(X) :-- M(p)(Ix).<br />

These constants are called <strong>the</strong> p-convexity and p-concavity constants <strong>of</strong> X. So X is p-<br />

convex (p-concave) if and only if X* is p*-concave (p*-convex).<br />

A p-convex and r-concave <strong>Banach</strong> lattice can be renormed with an equivalent lattice<br />

norm so that <strong>the</strong> p-convexity and r-concavity constants are both one [ 15, 1.d.8]. In particular,<br />

a lattice which is both p-convex and p-concave is lattice isomorphic to an abstract<br />

Lp space.<br />

If T : X --+ Y is a positive operator between <strong>Banach</strong> lattices it is easy to check (see [ 15,<br />

1.d.9]) <strong>the</strong> <strong>in</strong>equality<br />

(i ITx, I p<br />

n=l<br />

28 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

By p-concavity, this last quantity is dom<strong>in</strong>ated by<br />

p) lip<br />

M(p)(X)<br />

M(p) (X) Bp Ixnl 2<br />

1/2<br />

This gives <strong>the</strong> equivalence <strong>of</strong> <strong>the</strong> norm <strong>of</strong> <strong>the</strong> square function <strong>of</strong> X l .... , Xn with a certa<strong>in</strong><br />

Rademacher average:<br />

(5)<br />

p) 1/p<br />

A1 Ixnl 2<br />

n=l<br />

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 29<br />

Let {y*}neC__l be <strong>the</strong> functionals <strong>in</strong> Y* biorthogonal to {Yn}nCC=l. By compos<strong>in</strong>g S with (contractive)<br />

projections onto <strong>the</strong> span <strong>of</strong> <strong>in</strong>itial segments <strong>of</strong> {Yn}oc<br />

n=l' it can be assumed that<br />

Y is f<strong>in</strong>ite dimensional, <strong>in</strong> which case {Yn*} is a monotonely unconditional basis for Y*.<br />

To prove <strong>the</strong> right <strong>in</strong>equality <strong>in</strong> (12), given y~nN=l OtnYn, choose a norm one functional<br />

ZnL 1/3nyn , <strong>in</strong> y, so that II~--~=l N ot~y~ II- Y~'~=l/~ N y,( ~ ~-~=l N ot, y~). Us<strong>in</strong>g (6) and <strong>the</strong> left<br />

<strong>in</strong>equality <strong>in</strong> (10), we <strong>the</strong>n get<br />

N<br />

• Otn Yn -- Z 13notnS* y* (Tyn)<br />

n=l<br />

n--1<br />

30 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

unconditional basis for a complemented subspace <strong>of</strong> an L1 (/Z) space is equivalent to <strong>the</strong><br />

unit vector basis for g~l. S<strong>in</strong>ce C(0, 1) and L 1(0, 1) conta<strong>in</strong> subspaces isomorphic to ~2<br />

while co and ~1 do not, this gives ano<strong>the</strong>r pro<strong>of</strong> that nei<strong>the</strong>r C (0, 1) nor L 1(0, 1) has an<br />

unconditional basis.<br />

By us<strong>in</strong>g a p-convexification procedure, it is possible to build a scale <strong>of</strong> <strong>Banach</strong> lattices<br />

start<strong>in</strong>g with a lattice X <strong>in</strong> a manner analogous to how <strong>the</strong> scale <strong>of</strong> L p(/z) spaces<br />

is constructed from L1 (/Z). For simplicity, we assume that X is a lattice <strong>of</strong>/z-measurable<br />

functions (for <strong>the</strong> general case see [15, p. 53]). For 1 < p < oc, <strong>the</strong> p-convexification <strong>of</strong><br />

X is <strong>the</strong> space X (p) <strong>of</strong> all/z-measurable functions x for which IxlPsign(x) is <strong>in</strong> X. This is<br />

easily seen to be a <strong>Banach</strong> lattice under <strong>the</strong> norm<br />

. - It<br />

1/p<br />

ILx 9<br />

The space X (p) is p-convex with M (p) (X (p)) -- 1. More generally, if X is r-convex and s-<br />

concave, <strong>the</strong>n X (p) is pr-convex and ps-concave with moduli M (pr) (X (p)) ~ M (r) (X) 1/p<br />

and M(ps)(X (p)) O, 1 + tx*(y) = x*(x + ty)

- - p(r)<br />

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 31<br />

By us<strong>in</strong>g <strong>the</strong> triangle <strong>in</strong>equality one checks that <strong>the</strong> function Ilx+tyll-Ilxllt is an <strong>in</strong>creas<strong>in</strong>g<br />

function <strong>of</strong> t on (0, ec) and thus<br />

Similarly,<br />

x*(y) lim<br />

t-+O+ -t<br />

IIx II<br />

Thus if <strong>the</strong>se two limits co<strong>in</strong>cide, <strong>the</strong> value <strong>of</strong> x* (y) is uniquely determ<strong>in</strong>ed. On <strong>the</strong> o<strong>the</strong>r<br />

hand, if <strong>the</strong>se two limits differ, it follows from <strong>the</strong> Hahn-<strong>Banach</strong> <strong>the</strong>orem that for any X<br />

satisfy<strong>in</strong>g<br />

IIx - ty II - IIx II IIx + ty II - IIx II<br />

lim<br />

~ )~ 0 <strong>the</strong>re exists 6x(e) = 3(E) > 0 so that<br />

sup{ II (x + y)/21l Ilxll- Ilyll- 1; IIx - yll- ~} - 1 - 6(e).<br />

The function 6x(e) is called <strong>the</strong> modulus <strong>of</strong> convexity <strong>of</strong> X. It is geometrically obvious (and<br />

even true) that <strong>in</strong> <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> 3(e), <strong>the</strong> equalities can be replaced by <strong>the</strong> <strong>in</strong>equalities<br />

Ilxll ~< 1, Ilyll ~< 1, IIx - yll/> ~ without chang<strong>in</strong>g <strong>the</strong> value <strong>of</strong> 6(e).<br />

One <strong>of</strong> <strong>the</strong> first <strong>the</strong>orems to relate <strong>the</strong> geometry <strong>of</strong> <strong>the</strong> norm to l<strong>in</strong>ear topological properties<br />

is <strong>the</strong> follow<strong>in</strong>g. A uniformly convex space is reflexive. In order to see this, we may<br />

assume as well that X is separable. If X is not reflexive, <strong>the</strong>n for any )~ > 0 <strong>the</strong>re is x** <strong>in</strong><br />

<strong>the</strong> unit sphere <strong>of</strong> X** whose distance to X exceeds 1 - )~. Let A be a countable subset <strong>of</strong><br />

<strong>the</strong> unit sphere <strong>of</strong> X* which determ<strong>in</strong>es <strong>the</strong> norm <strong>of</strong> Y -- spanX U {x**}; that is, for each<br />

y** <strong>in</strong> Y, Ily**ll - sup{y**(x*)" x* 6 A}. Let {Xn}neC__l be a sequence <strong>in</strong> <strong>the</strong> unit sphere <strong>of</strong><br />

X so that x*(xn) --+ x**(x*) for every x* <strong>in</strong> A. Then limn,m-+,c Ilxn + xmll -- 2, while for<br />

every n, lim<strong>in</strong>fm-+ec Ilxn -- Xm II ~> 1 - )~. This argument shows that for nonrettexive X,<br />

~x(e) = 0 for every 0 < e < 1 and <strong>in</strong> particular that X is not uniformly convex.<br />

A <strong>Banach</strong> space X is said to be uniformly smooth if <strong>the</strong> function<br />

--sup/IIx + rYll + Ilx - ryll<br />

px(r)<br />

/ 2<br />

- 1. IIx II - Ily II - 1 }<br />

satisfies p(r) = o(r) as r --+ 0. Aga<strong>in</strong> one checks easily that a uniformly smooth <strong>Banach</strong><br />

space is reflexive.

32 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

There is a complete duality between uniform convexity and uniform smoothness. The<br />

space X is uniformly convex if and only if X* is uniformly smooth. This follows from a<br />

formula which connects <strong>the</strong> moduli 8x and Px*"<br />

px, (r) -- sup{rE/2 - 8x(~)" 0 sup{rE/2 - 8x(~): 0 ~< ~ ~< 2}. Conversely, given r > 0 and x*,<br />

y* <strong>in</strong> <strong>the</strong> unit sphere <strong>of</strong> X*, take x and y <strong>in</strong> <strong>the</strong> unit sphere <strong>of</strong> X so that (x* + ry*)(x) =<br />

IIx* + rY*ll and (x* - ry*)(x) = IIx* - ry*ll (if one wishes to avoid reflexivity at this stage<br />

one can choose x and y where <strong>the</strong> norms <strong>of</strong> <strong>the</strong> functionals are almost atta<strong>in</strong>ed). Then<br />

IIx* + ry*ll + IIx* - ry*ll ~< x*(x + y) + ry*(x - y)<br />

IIx + yll + rllx - yll - 28x(~) + ~r,<br />

where E - IIx - y II. This proves <strong>the</strong> reverse <strong>in</strong>equality.<br />

The moduli <strong>of</strong> convexity and smoothness <strong>of</strong> a Hilbert space H can be easily computed<br />

from <strong>the</strong> parallelogram identity:<br />

pH(r)<br />

E2 E 2<br />

1 = )<br />

4 8<br />

-- v/1 + "r 2- 1 = r<br />

T +<br />

2<br />

Hilbert spaces are <strong>the</strong> "most uniformly convex" and "most uniformly smooth" spaces <strong>in</strong><br />

<strong>the</strong> sense that if X is any space whose dimension is at least two, <strong>the</strong>n 8/-/(E)/> 8x(E) and<br />

p/-/(r) ~< px(r). For <strong>in</strong>f<strong>in</strong>ite dimensional X this is an immediate consequence <strong>of</strong> Dvoretzky's<br />

<strong>the</strong>orem, to be discussed <strong>in</strong> Section 8, but a direct 2-dimensional geometrical argument<br />

gives <strong>the</strong> general case.<br />

The moduli <strong>of</strong> convexity and smoothness <strong>of</strong> <strong>the</strong> L p spaces can be computed exactly. It<br />

is somewhat easier to describe <strong>the</strong>ir asymptotic behavior near zero, and this is what really<br />

matters for <strong>Banach</strong> space <strong>the</strong>ory. The result is<br />

E 2<br />

(p- 1)--~- + o(e2), if 1 < p ~< 2,<br />

p2P '<br />

if2~

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 33<br />

z'P jr_ O(rP),<br />

p - _<br />

IO L p ( r: ) -- 7:2<br />

ifl < p~ 0 (see [15, If.l]).<br />

S<strong>in</strong>ce it is easier to do analysis on a <strong>Banach</strong> space which has a norm with good geometric<br />

properties than on a general space, it is important to know which <strong>Banach</strong> spaces can<br />

be equivalently renormed so as to become strictly convex or smooth or uniformly convex<br />

or uniformly smooth, and it is useful to have <strong>in</strong> <strong>the</strong>se last cases moduli which are as good<br />

as possible. The simplest, but never<strong>the</strong>less useful, renorm<strong>in</strong>g technique is as follows. Suppose<br />

that T is an <strong>in</strong>jective operator from a <strong>Banach</strong> space X <strong>in</strong>to a strictly convex <strong>Banach</strong><br />

space Y. Then it is easy to check that IIIxlll := Ilxll + IITxll is an equivalent strictly convex<br />

norm on X. Moreover, if T is an isomorphism and Y is uniformly convex, <strong>the</strong>n II1" III is an<br />

equivalent uniformly convex norm on X. Now if X is separable, <strong>the</strong>n <strong>the</strong>re is an <strong>in</strong>jective<br />

operator T from X <strong>in</strong>to ~2, so Ilxlll = IIx II + IITx II is an equivalent strictly convex norm<br />

on X. Also, <strong>the</strong>re is an operator S from ~2 <strong>in</strong>to X with dense range, so S* is an <strong>in</strong>jective<br />

operator from X* <strong>in</strong>to ~ --~2 and IIx*l12- IIx*ll + IIS*x*ll def<strong>in</strong>es an equivalent strictly<br />

convex norm on X*. S<strong>in</strong>ce <strong>the</strong> adjo<strong>in</strong>t operator S* is weak* to weak cont<strong>in</strong>uous, I1" 112 is<br />

dual to a (necessarily smooth) norm I1" 112 on X. If X is smooth and T is an <strong>in</strong>jective operator<br />

from X <strong>in</strong>to ~2 <strong>the</strong>n (llx II 2 + II Zx II 2) ~/2 is an equivalent norm which is both strictly<br />

convex and smooth. Hence every separable <strong>Banach</strong> space has an equivalent norm which<br />

is both strictly convex and smooth.<br />

For certa<strong>in</strong> nonseparable spaces; <strong>in</strong> particular, ~ (F) with F uncountable (see [6, Chapter<br />

II.7]), <strong>the</strong>re may be no equivalent strictly convex or smooth norm.<br />

If we are <strong>in</strong>terested <strong>in</strong> obta<strong>in</strong><strong>in</strong>g a uniformly convex or smooth equivalent norm we<br />

have to restrict attention to reflexive spaces. However, not every reflexive space can be so<br />

OO /7 .<br />

renormed. For example, if ~,=1 g l )2 had an equivalent uniformly convex norm II II and<br />

I1" II, denotes <strong>the</strong> restriction <strong>of</strong> I1" Ilto <strong>the</strong> nth coord<strong>in</strong>ate space ~, <strong>the</strong>n <strong>the</strong> expression<br />

IIIx III := lim IIx II, (where "lim" is <strong>in</strong>terpreted to be a limit over some free ultrafilter on N or<br />

a <strong>Banach</strong> limit or <strong>the</strong> limit along an appropriate subsequence) def<strong>in</strong>es an equivalent norm<br />

on <strong>the</strong> f<strong>in</strong>itely supported vectors <strong>in</strong> el which extends uniquely to an equivalent uniformly<br />

convex norm on ~1, but this is impossible.<br />

There is a characterization <strong>of</strong> those spaces on which <strong>the</strong>re is an equivalent uniformly<br />

convex norm. These spaces, called superreflexive spaces, are discussed <strong>in</strong> Section 9. The<br />

superreflexive spaces are also <strong>the</strong> class <strong>of</strong> spaces on which <strong>the</strong>re is an equivalent uniformly<br />

smooth norm. (These deep facts are discussed <strong>in</strong> [29].) If a space X has an equivalent<br />

uniformly convex norm II1" III, <strong>the</strong>n <strong>the</strong> equivalent uniformly convex norms are dense <strong>in</strong> <strong>the</strong><br />

metric space <strong>of</strong> equivalent norms (considered as bounded functions on <strong>the</strong> unit sphere <strong>of</strong><br />

X). Take, for example, I1" II +E II1"111. The uniformly convex equivalent norms also form a G~<br />

set s<strong>in</strong>ce those whose modulus <strong>of</strong> convexity at 1/n is positive forms an open set. Thus <strong>the</strong><br />

equivalent uniformly convex norms on X is a dense G~ <strong>in</strong> <strong>the</strong> space <strong>of</strong> equivalent norms<br />

on X. S<strong>in</strong>ce, as we mentioned, X* also admits an equivalent uniformly convex norm, it

34 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

follows by duality that <strong>the</strong> equivalent uniformly smooth norms on X is also a dense G~ <strong>in</strong><br />

<strong>the</strong> space <strong>of</strong> equivalent norms on X, hence so is <strong>the</strong> family <strong>of</strong> equivalent norms which are<br />

simultaneously uniformly convex and uniformly smooth.<br />

In Section 8 it is po<strong>in</strong>ted out that an <strong>in</strong>f<strong>in</strong>ite dimensional L p (#) space cannot be equivalently<br />

renormed so as to have a better modulus <strong>of</strong> convexity or smoothness than that <strong>of</strong> <strong>the</strong><br />

natural norm.<br />

There has long been a desire to describe reflexivity geometrically (that is, to show that a<br />

space is reflexive if and only if <strong>the</strong>re is an equivalent norm on <strong>the</strong> space that satisfies some<br />

geometrical condition), and <strong>the</strong> notion <strong>of</strong> uniform convexity pushed <strong>in</strong> that direction. This<br />

problem was recently solved (at least for separable spaces) and is discussed <strong>in</strong> [29].<br />

Early <strong>in</strong> a first course <strong>in</strong> functional analysis a student learns that a <strong>Banach</strong> space is reflexive<br />

if and only if its closed unit ball is weakly compact. There is a beautiful and useful<br />

characterization <strong>of</strong> weak compactness, called James' <strong>the</strong>orem, which does not explicitly<br />

<strong>in</strong>volve <strong>the</strong> weak topology: A nonempty closed convex subset C <strong>of</strong> a <strong>Banach</strong> space X is<br />

weakly compact if and only if every x* <strong>in</strong> X* atta<strong>in</strong>s its maximum on C. The only if part<br />

is <strong>of</strong> course trivial. For <strong>the</strong> hard direction see [11, Theorem 79] or [26] for an accessible<br />

pro<strong>of</strong> when X is separable. This <strong>the</strong>orem says that on a nonreflexive space X <strong>the</strong>re exist<br />

l<strong>in</strong>ear functionals which do not atta<strong>in</strong> <strong>the</strong>ir norm on <strong>the</strong> unit ball <strong>of</strong> X. Never<strong>the</strong>less, <strong>the</strong><br />

functionals which atta<strong>in</strong> <strong>the</strong>ir norm on <strong>the</strong> unit ball is a rich set: The Bishop-Phelps <strong>the</strong>orem<br />

says: Let C be a nonempty closed bounded subset <strong>of</strong> a <strong>Banach</strong> space X. Then <strong>the</strong><br />

functionals which atta<strong>in</strong> <strong>the</strong>ir maximum on C is (norm) dense <strong>in</strong> X*. This <strong>the</strong>orem, which<br />

is <strong>the</strong> start<strong>in</strong>g po<strong>in</strong>t <strong>of</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> optimization on <strong>Banach</strong> spaces, has many extensions<br />

and applications (see [25]).<br />

We outl<strong>in</strong>e <strong>the</strong> pro<strong>of</strong> <strong>of</strong> <strong>the</strong> Bishop-Phelps <strong>the</strong>orem. First note that if f is a cont<strong>in</strong>uous<br />

bounded function on a complete metric space (U, d), <strong>the</strong>n <strong>the</strong>re is, for every E > 0, a po<strong>in</strong>t<br />

u0 <strong>in</strong> U so that f(u) fl for (x, t) <strong>in</strong> Kl and<br />

u*(x) + at

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 35<br />

bounded closed convex set C <strong>in</strong> a complex <strong>Banach</strong> space X so that for every x* <strong>in</strong> X*, Ix*l<br />

does not atta<strong>in</strong> its maximum on C. This is discussed <strong>in</strong> [26].<br />

We conclude this section by mention<strong>in</strong>g <strong>the</strong> geometric mean<strong>in</strong>g <strong>of</strong> <strong>the</strong> Radon-Nikod3)m<br />

property or RNP, an analytical concept that will be discussed <strong>in</strong> Section 7. A slice <strong>of</strong><br />

a closed bounded convex set C is a set <strong>of</strong> <strong>the</strong> form S(C,x*,~) = {x E C: x*(x) >1<br />

SUpyec x*(y) - c~} with x* <strong>in</strong> X* and ot > 0. C is called dentable provided that for each<br />

E > 0 <strong>the</strong>re is a slice <strong>of</strong> C which has diameter smaller than ~. In [8, Th. V 3.9] and [3,<br />

Th. 5.8] it is shown that X has <strong>the</strong> RNP if and only if every nonempty closed bounded<br />

convex subset <strong>of</strong> X is dentable. Here we show how dentability can be used to derive o<strong>the</strong>r<br />

geometric conditions, <strong>in</strong> particular, a version <strong>of</strong> <strong>the</strong> Kre<strong>in</strong>-Milman <strong>the</strong>orem valid for noncompact<br />

subsets <strong>of</strong> a space which has <strong>the</strong> RNE If X has <strong>the</strong> RNP, <strong>the</strong>n every closed bounded<br />

convex subset C <strong>of</strong> X is <strong>the</strong> closed convex hull <strong>of</strong> its extreme po<strong>in</strong>ts. Recall that a face <strong>of</strong> a<br />

convex set C is a nonempty (necessarily convex) subset F such that if )~x + (1 - ;~)y E F<br />

with x, y <strong>in</strong> C and 0 < )~ < 1 <strong>the</strong>n x and y are <strong>in</strong> F. Extreme po<strong>in</strong>ts are faces consist<strong>in</strong>g<br />

<strong>of</strong> a s<strong>in</strong>gle po<strong>in</strong>t. From <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> slice it is obvious that if I(x* - Y*)(Y) I ~< 3 for all<br />

y <strong>in</strong> C <strong>the</strong>n S(C, y*, ~ - 23) C S(C, x*, ~) as long as ot > 23. We first show that C has<br />

an extreme po<strong>in</strong>t. Take any slice S(C, x*, c~) <strong>of</strong> C whose diameter is less than one. By <strong>the</strong><br />

observation above and <strong>the</strong> Bishop-Phelps <strong>the</strong>orem, <strong>the</strong>re is a y* arbitrarily near x* which<br />

atta<strong>in</strong>s a maximum on C and so that <strong>the</strong> po<strong>in</strong>ts P1 <strong>in</strong> C at which y* atta<strong>in</strong>s its maximum<br />

is conta<strong>in</strong>ed <strong>in</strong> <strong>the</strong> slice S(C, x*, ~) and thus has diameter less than one. Evidently P1 is<br />

a closed face <strong>of</strong> C and <strong>of</strong> course is dentable s<strong>in</strong>ce X has <strong>the</strong> RNE By <strong>in</strong>duction we get a<br />

sequence {Pn}~~ so that Pn+l is a face <strong>of</strong> Pn and diamPn ~< 1/n. S<strong>in</strong>ce a face <strong>of</strong> a face<br />

is a face, all <strong>the</strong> Pn's are faces <strong>of</strong> C and <strong>the</strong>ir <strong>in</strong>tersection is an extreme po<strong>in</strong>t <strong>of</strong> C which<br />

is <strong>in</strong> <strong>the</strong> slice S(C, x*, ~). To f<strong>in</strong>ish <strong>the</strong> pro<strong>of</strong>, let K be <strong>the</strong> closed convex hull <strong>of</strong> <strong>the</strong> extreme<br />

po<strong>in</strong>ts <strong>of</strong> C. If K were properly conta<strong>in</strong>ed <strong>in</strong> C, <strong>the</strong>n <strong>the</strong> separation <strong>the</strong>orem and <strong>the</strong><br />

Bishop-Phelps <strong>the</strong>orem would yield a functional x* which atta<strong>in</strong>s a maximum on C and so<br />

that <strong>the</strong> set <strong>of</strong> po<strong>in</strong>ts P <strong>in</strong> C at which x* atta<strong>in</strong>s its maximum is disjo<strong>in</strong>t from K. By what<br />

we have proved P has an extreme po<strong>in</strong>t which is afortiori an extreme po<strong>in</strong>t <strong>of</strong> C, which<br />

contradicts <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> K.<br />

It is open whe<strong>the</strong>r this extreme po<strong>in</strong>t property for every nonempty closed bounded convex<br />

subset <strong>of</strong> a space actually characterizes spaces with <strong>the</strong> RNE The statement about extreme<br />

po<strong>in</strong>ts is however valid <strong>in</strong> a stronger form which trivially implies that <strong>the</strong> dentability<br />

condition is equivalent to <strong>the</strong> RNE A po<strong>in</strong>t x <strong>in</strong> C is called a exposed po<strong>in</strong>t <strong>of</strong> C if <strong>the</strong>re is<br />

x* <strong>in</strong> X* which atta<strong>in</strong>s its maximum on C exactly at <strong>the</strong> s<strong>in</strong>gle po<strong>in</strong>t x. If also <strong>the</strong> diameter<br />

<strong>of</strong> <strong>the</strong> slice S(C, x*, c~) tends to zero as ot --+ 0+, <strong>the</strong>n x is called a strongly exposed po<strong>in</strong>t<br />

<strong>of</strong> C. For example, every po<strong>in</strong>t on <strong>the</strong> unit sphere <strong>of</strong> a uniformly convex space is a strongly<br />

exposed po<strong>in</strong>t <strong>of</strong> <strong>the</strong> unit ball. The stronger statement to which we alluded above is: Every<br />

nonempty closed bounded convex subset <strong>of</strong> a space with <strong>the</strong> RNP is <strong>the</strong> closed convex hull<br />

<strong>of</strong> its strongly exposed po<strong>in</strong>ts.<br />

7. Analysis <strong>in</strong> <strong>Banach</strong> spaces<br />

In this section we describe <strong>the</strong> basic facts concern<strong>in</strong>g <strong>in</strong>tegration and differentiation <strong>in</strong><br />

<strong>Banach</strong> spaces as well as <strong>the</strong> connection <strong>of</strong> <strong>the</strong>se topics to convexity. Our "po<strong>in</strong>t <strong>of</strong> view"

36 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

is to take as known <strong>the</strong> scalar <strong>the</strong>ory but not assume that <strong>the</strong> reader has any familiarity with<br />

<strong>the</strong> vector valued <strong>the</strong>ory. Much <strong>of</strong> what we treat <strong>in</strong> this section is conta<strong>in</strong>ed ei<strong>the</strong>r <strong>in</strong> [8] or<br />

<strong>in</strong> [ 11,12]. Almost everyth<strong>in</strong>g is <strong>in</strong> [3].<br />

Let (s #) be a complete a-f<strong>in</strong>ite measure space and X a <strong>Banach</strong> space. A function<br />

from s --~ X <strong>of</strong> <strong>the</strong> form ~<strong>in</strong>=l xi 1 ai with each Ai a measurable subset <strong>of</strong> s is called a<br />

simple function. A function f : ~2 --+ X is called strongly measurable or just measurable<br />

if it is <strong>the</strong> limit almost everywhere <strong>of</strong> a sequence <strong>of</strong> simple functions. A function f is<br />

called scalarly measurable provided x*f is measurable for each x* <strong>in</strong> X*. A function<br />

f is measurable if and only if it is scalarly measurable and <strong>the</strong> range <strong>of</strong> f is essentially<br />

separable, which means that for some set A <strong>of</strong> measure zero, f[X2 ~ A] is separable. The<br />

only if assertion is clear. To verify <strong>the</strong> if part, we may assume that X is separable and<br />

#<br />

select a sequence ~x*/cc <strong>in</strong> <strong>the</strong> unit ball <strong>of</strong> X* so that Ilxll - SUPn Xn (X) for every x <strong>in</strong><br />

t n "n=l<br />

X. Thus for each fixed x <strong>in</strong> X, <strong>the</strong> function II f - x II is a measurable real valued function<br />

on Y2. S<strong>in</strong>ce X is separable, given e > 0 <strong>the</strong>re is a sequence {xn}~ec__ 1 <strong>in</strong> X and a sequence<br />

{An}n~l <strong>of</strong> disjo<strong>in</strong>t measurable subsets <strong>of</strong> s so that Ilf- ~nXnlAnll < ~. That is, f can<br />

be approximated uniformly by functions tak<strong>in</strong>g only countably many values and hav<strong>in</strong>g all<br />

level sets measurable and hence f is <strong>the</strong> limit a.e. <strong>of</strong> a sequence <strong>of</strong> simple functions.<br />

We next def<strong>in</strong>e <strong>the</strong> notion <strong>of</strong> <strong>the</strong> Bochner <strong>in</strong>tegral <strong>of</strong> a vector valued function. This is <strong>the</strong><br />

"strongest" <strong>of</strong> <strong>the</strong> various vector valued <strong>in</strong>tegrals which have been considered and is <strong>the</strong><br />

one most useful for <strong>the</strong> topics treated <strong>in</strong> this Handbook.<br />

If f is a simple function supported on a set <strong>of</strong> f<strong>in</strong>ite measure, def<strong>in</strong>e f f d/x =<br />

~x~X Iz[f = x]x. This is <strong>of</strong> course a f<strong>in</strong>ite sum. It is easy to see directly, and also follows<br />

from <strong>the</strong> scalar case by compos<strong>in</strong>g with l<strong>in</strong>ear functionals, that <strong>the</strong> <strong>in</strong>tegral is l<strong>in</strong>ear<br />

on this class <strong>of</strong> simple functions. If f is <strong>the</strong> a.e. limit <strong>of</strong> a sequence {fn }n~_-i <strong>of</strong> simple functions<br />

supported on sets <strong>of</strong> f<strong>in</strong>ite measure and limn,m f IlL - fm II d# = 0, <strong>the</strong>n f f d# is<br />

def<strong>in</strong>ed to be limn f f~ d~. It is easy to check that f f d# does not depend on <strong>the</strong> particular<br />

sequence {fn }n~_-i and that f f d# exists if and only if f is measurable and f Ilfll d# < oe.<br />

That <strong>the</strong> <strong>in</strong>tegral has <strong>the</strong> expected properties ei<strong>the</strong>r follows from <strong>the</strong> scalar case by compos<strong>in</strong>g<br />

with l<strong>in</strong>ear functionals or is easy to check directly; see [8, pp. 44-52]. In particular,<br />

II f f d# II ~< f II f II d# and T f f dlz = f Tf d# whenever f is <strong>in</strong>tegrable and T is an operator.<br />

That <strong>the</strong> usual differentiability properties hold for <strong>the</strong> Bochner <strong>in</strong>tegral can be deduced<br />

from <strong>the</strong> scalar <strong>the</strong>orems even without recall<strong>in</strong>g <strong>the</strong> pro<strong>of</strong>s <strong>in</strong> that case. Suppose that f is<br />

a Lebesgue <strong>in</strong>tegrable X valued function on R n . Then for a.e. u <strong>in</strong> R n, we have<br />

lim m(B(O, r)) -1<br />

r--+O (u,r) Iif(v) - f(u)II dv = 0 (13)<br />

and thus also f(u) - limr~om(B(O, r)) -1 fB(u,r) f(v) dv (here <strong>in</strong>tegration is with respect<br />

to Lebesgue measure m on Rn). Indeed, we may assume that X is separable and {Xn}n~<br />

is dense <strong>in</strong> X. Then by <strong>the</strong> scalar <strong>the</strong>orem,<br />

lim m(B(O, r)) -1 s<br />

r--+O<br />

II f(v) - xnll dv = IIf(u) - Xnll<br />

(u,r)

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 37<br />

for a.e. u and every n. For a po<strong>in</strong>t u for which this holds for every n we have<br />

limsupm(B(O, r)) -1 fB<br />

r--+O<br />

(u,r)<br />

fll<br />

lif(v) - f(u)ll dv<br />

38 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

Chapter 3]) is that a function f from <strong>the</strong> complex plane <strong>in</strong>to Y is analytic if y* f is analytic<br />

for each y* <strong>in</strong> Y*.<br />

One important <strong>Banach</strong> space <strong>the</strong>ory property that comes from differentiation <strong>the</strong>ory is<br />

<strong>the</strong> Radon-Nikodjm property or RNP. One way to def<strong>in</strong>e this notion is <strong>the</strong> follow<strong>in</strong>g: The<br />

space X has <strong>the</strong> RNP if every Lipschitz function from N <strong>in</strong>to X is differentiable a.e. Here<br />

we do not need to specify G~teaux or Fr6chet differentiability s<strong>in</strong>ce <strong>the</strong> notions obviously<br />

co<strong>in</strong>cide for all functions from R. The RNP is usually def<strong>in</strong>ed by <strong>the</strong> requirement that<br />

<strong>the</strong> Radon-Nikod~m <strong>the</strong>orem holds for X valued measures <strong>of</strong> f<strong>in</strong>ite total variation (see,<br />

for example, [8]). We shall discuss that aspect and o<strong>the</strong>r equivalences <strong>of</strong> <strong>the</strong> RNP later <strong>in</strong><br />

this section. Here just note that a <strong>Banach</strong> space valued Lipschitz function f on [0, 1 ] has<br />

separable range, so its a.e. derivative fl, if it exists, is also separably valued and thus is a<br />

measurable function s<strong>in</strong>ce x*f is measurable for each x* <strong>in</strong> X*.<br />

First let us get a feel<strong>in</strong>g for which spaces have <strong>the</strong> RNP. It is clear from <strong>the</strong> def<strong>in</strong>ition<br />

that a subspace <strong>of</strong> a space with <strong>the</strong> RNP has <strong>the</strong> RNP and that a space all <strong>of</strong> whose<br />

separable subspaces have <strong>the</strong> RNP has <strong>the</strong> RNP as well. The mapp<strong>in</strong>g t w-~ l(0,t) from<br />

(0, 1) <strong>in</strong>to L 1(0, 1) shows that L 1(0, 1) fails <strong>the</strong> RNP. Also co fails <strong>the</strong> RNP. This is seen<br />

by consider<strong>in</strong>g <strong>the</strong> function f'R --+ co @~ co def<strong>in</strong>ed by f(t) - {fo s<strong>in</strong>nsds}n~--1 9<br />

{fo cosns ds}n~__l . That f maps <strong>in</strong>to co follows from <strong>the</strong> Riemann-Lebesgue lemma but<br />

<strong>the</strong> only possible candidate for a derivative is {s<strong>in</strong>ns}n~__l 9 {cosns}n~__l which is not <strong>in</strong> co<br />

for any s.<br />

Separable conjugate spaces have <strong>the</strong> RNP and thus all reflexive spaces have <strong>the</strong> RNP<br />

(<strong>in</strong>cidentally, this yields ano<strong>the</strong>r pro<strong>of</strong> <strong>of</strong> <strong>the</strong> fact mentioned <strong>in</strong> Section 3 that L 1 (0, 1) does<br />

not embed <strong>in</strong>to a separable conjugate space). To see this, let f be a Lipschitz function <strong>in</strong>to<br />

a separable conjugate space Z* and let {Zn }ne~=l be dense <strong>in</strong> Z. For all n, <strong>the</strong> scalar Lipschitz<br />

function f(t)(Zn) is differentiable for a.e.t. At a po<strong>in</strong>t to where all <strong>of</strong> <strong>the</strong>se functions are<br />

differentiable, f(to)(z) is differentiable for every z <strong>in</strong> Z (observe that h-l(f(to -+- h) -<br />

f (to))(z) - k -1 (f (to + k) - f (to))(z) --+ 0 as h, k --+ 0 because <strong>the</strong> difference quotient is<br />

uniformly bounded s<strong>in</strong>ce f is Lipschitz and tends to zero on <strong>the</strong> dense set {Zn }n~=l). From<br />

this we conclude that <strong>the</strong> limit g(t) := limh__,oh-l(f(t + h) - f(t)) exists a.e., but <strong>the</strong><br />

limit is only <strong>in</strong> <strong>the</strong> weak* sense. This is all that can be said us<strong>in</strong>g just <strong>the</strong> separability <strong>of</strong> Z.<br />

However, s<strong>in</strong>ce Z* is separable, we deduce that g is measurable (g has separable range and<br />

Jig(t) - z* II is clearly measurable for every z* <strong>in</strong> Z*; this is all that was used <strong>in</strong> <strong>the</strong> pro<strong>of</strong> <strong>of</strong><br />

measurability <strong>in</strong> <strong>the</strong> beg<strong>in</strong>n<strong>in</strong>g <strong>of</strong> this section). Also g is bounded by <strong>the</strong> Lipschitz constant<br />

<strong>of</strong> f, so <strong>the</strong> Bochner <strong>in</strong>tegral G(s) "- fo g(t)dt is well def<strong>in</strong>ed for all s. Evaluat<strong>in</strong>g both<br />

sides at an arbitrary z <strong>in</strong> Z we see from <strong>the</strong> scalar <strong>the</strong>ory that G(s)(z) = (f(s) - f(0))(z)<br />

so that f(s) -- G(s) + f(O) and hence by what we proved on <strong>the</strong> differentiation <strong>of</strong> <strong>the</strong><br />

<strong>in</strong>tegral we conclude that f'(s) = G'(s) = g(s) a.e. also <strong>in</strong> <strong>the</strong> sense that Ilh -1 (f(s + h) -<br />

f(s)) - g(s)ll ~ 0 as h -+ 0.<br />

Thus every subspace <strong>of</strong> a separable conjugate space has <strong>the</strong> RNP and a space with <strong>the</strong><br />

RNP has no subspace isomorphic to co or L 1(0, 1). These facts give useful criteria for<br />

<strong>the</strong> RNP and <strong>in</strong> certa<strong>in</strong> cases, such as spaces with an unconditional basis, allow a complete<br />

determ<strong>in</strong>ation whe<strong>the</strong>r a space has <strong>the</strong> RNR However, <strong>the</strong>re do exist separable spaces<br />

with <strong>the</strong> RNP which do not embed <strong>in</strong>to a separable conjugate space and <strong>the</strong>re are spaces<br />

fail<strong>in</strong>g <strong>the</strong> RNP which do not conta<strong>in</strong> isomorphic copies <strong>of</strong> ei<strong>the</strong>r co or L 1(0, 1) (see [3,<br />

Section 3.4]).

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 39<br />

Before stat<strong>in</strong>g <strong>the</strong> usual def<strong>in</strong>ition <strong>of</strong> <strong>the</strong> RNP we need to <strong>in</strong>troduce <strong>the</strong> notion <strong>of</strong> what<br />

is termed <strong>in</strong> [8] a "countably additive vector measure on a a-algebra". If X is a <strong>Banach</strong><br />

space, an X valued measure r is a countably additive function from a a-algebra <strong>in</strong>to X. For<br />

this to make sense, <strong>the</strong> series y~ r(A,) must converge unconditionally for each sequence<br />

{An}n~__l <strong>of</strong> disjo<strong>in</strong>t sets <strong>in</strong> <strong>the</strong> o--algebra. One def<strong>in</strong>es <strong>the</strong> total variation It] <strong>of</strong> an X valued<br />

measure by ]r ](A) = sup ~n lit (A,)11, where <strong>the</strong> supremum is over all partitions <strong>of</strong> A <strong>in</strong>to<br />

f<strong>in</strong>itely many (or countably many; it is <strong>the</strong> same) disjo<strong>in</strong>t sets <strong>in</strong> <strong>the</strong> cr-algebra (s The<br />

measure r is said to be <strong>of</strong>f<strong>in</strong>ite variation provided ]r 1(s < cx~. If r is <strong>of</strong> f<strong>in</strong>ite variation,<br />

<strong>the</strong>n It] is a f<strong>in</strong>ite scalar measure. For example, if s = 1~,/3 is <strong>the</strong> collection <strong>of</strong> all subsets<br />

<strong>of</strong> I~t, and {x,},~__l is a sequence <strong>in</strong> <strong>the</strong> <strong>Banach</strong> space X, <strong>the</strong>n <strong>the</strong> assignment r{n} "- Xn<br />

extends to an X valued measure on/3 if and only if ~n x. converges unconditionally. The<br />

measure is <strong>the</strong>n <strong>of</strong> f<strong>in</strong>ite variation if and only if ~.n ]]Xn 11 < c~. An example <strong>of</strong> an L p (0, 1),<br />

1 ~< p < cxz, valued measure is gotten by tak<strong>in</strong>g <strong>the</strong> Lebesgue measurable subsets <strong>of</strong> [0, 1 ]<br />

and def<strong>in</strong><strong>in</strong>g r(A) = 1a. This measure has f<strong>in</strong>ite variation only if p = 1.<br />

The <strong>the</strong>ory <strong>of</strong> X valued measures and more general vector measures is exposed <strong>in</strong> [8].<br />

In addition to <strong>the</strong>ir importance for <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces, vector measures are<br />

centrally important for spectral <strong>the</strong>ory; <strong>the</strong> ones encountered <strong>the</strong>re typically do not have<br />

f<strong>in</strong>ite variation.<br />

An X valued measure r is absolutely cont<strong>in</strong>uous with respect to <strong>the</strong>f<strong>in</strong>ite scalar measure<br />

# provided r (A) = 0 for every set <strong>of</strong> # measure zero. Because # is f<strong>in</strong>ite, this is equivalent<br />

to say<strong>in</strong>g that for every ~ > 0 <strong>the</strong>re is ~ > 0 so that Ilr(A)]l < ~ whenever #(A) < 6 (see<br />

[8, p. 10]). If r is an X valued measure <strong>of</strong> f<strong>in</strong>ite variation it is clear that r is absolutely<br />

cont<strong>in</strong>uous with respect to its total variation It] and even satisfies (14) below (with # :--<br />

Irl).<br />

An X valued measure r is differentiable with respect to a scalar measure lz provided that<br />

<strong>the</strong>re is an X valued measurable function g so that r(A) = fa g d# for every measurable set<br />

A. We say that <strong>the</strong> Radon-Nikodym <strong>the</strong>orem holds <strong>in</strong> X provided that if r is an X valued<br />

measure <strong>of</strong> f<strong>in</strong>ite variation and r is absolutely cont<strong>in</strong>uous with respect to a f<strong>in</strong>ite scalar<br />

measure #, <strong>the</strong>n r is differentiable with respect to #. If X satisfies this condition only for<br />

all separable f<strong>in</strong>ite scalar measures, we say that <strong>the</strong> separable Radon-Nikodym <strong>the</strong>orem<br />

holds <strong>in</strong> X (a measure # is called separable provided L1 (#) is separable). The usual<br />

def<strong>in</strong>ition is that a <strong>Banach</strong> space X has <strong>the</strong> RNP provided <strong>the</strong> Radon-Nikodym <strong>the</strong>orem<br />

holds <strong>in</strong> X and this is equivalent to say<strong>in</strong>g that <strong>the</strong> separable Radon-Nikod3~m <strong>the</strong>orem<br />

holds <strong>in</strong> X (see [8, Chapter III]). Later we prove this equivalence for separable X, but first<br />

we show a general space X has <strong>the</strong> RNP if and only if <strong>the</strong> separable Radon-NikodSm<br />

<strong>the</strong>orem holds <strong>in</strong> X.<br />

Suppose that <strong>the</strong> separable Radon-Nikod3)m <strong>the</strong>orem holds <strong>in</strong> X and let f'[0, 1 ] ~ X<br />

be a Lipschitz function. One def<strong>in</strong>es a l<strong>in</strong>ear mapp<strong>in</strong>g T from <strong>the</strong> step functions on [0, 1]<br />

<strong>in</strong>to X by sett<strong>in</strong>g Tl[a,b] := f(b) - f(a) for a sub<strong>in</strong>terval <strong>of</strong> [0, 1] and extend<strong>in</strong>g l<strong>in</strong>early.<br />

S<strong>in</strong>ce f is a Lipschitz function, <strong>the</strong> mapp<strong>in</strong>g T is cont<strong>in</strong>uous when <strong>the</strong> step functions are<br />

given <strong>the</strong> L 1(0, 1) norm, and hence T uniquely extends to an operator (also denoted by T)<br />

from L1 (0, 1) <strong>in</strong>to X. The assignment r(A) :-- T1A obviously def<strong>in</strong>es an X valued measure<br />

<strong>of</strong> f<strong>in</strong>ite variation which is absolutely cont<strong>in</strong>uous with respect to Lebesgue measure<br />

m, so we get an X valued measurable function g on [0, 1] for which r(A) = fag dm for<br />

every Lebesgue measurable subset <strong>of</strong> [0, 1 ]. In particular, f (t) -- f0 gdm + f (0) and thus

40 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

by what we proved <strong>in</strong> <strong>the</strong> beg<strong>in</strong>n<strong>in</strong>g <strong>of</strong> this section f'(t) exists a.e. on [0, 1 ] (and is equal<br />

to g(t)).<br />

Suppose that X has <strong>the</strong> RNE To see that <strong>the</strong> separable Radon-Nikod3~m <strong>the</strong>orem holds<br />

<strong>in</strong> X, suppose first that <strong>the</strong> scalar "control measure" Ix is Lebesgue measure on [0, 1] and<br />

that <strong>the</strong> X valued measure r satisfies<br />

[[r(A)I[ ~ 0 so that #(A) = fA f dv for every v-measurable set A. Of<br />

course, IX is <strong>the</strong>n also a separable measure and, as we have already remarked, r satisfies<br />

(14), so from what we already have proved <strong>the</strong>re is an X valued Ix-measurable function<br />

g so that r(A) = fAg dix for every Ix-measurable set A. Then f.g is v-measurable and<br />

r(A) = fA f" g dv for every v-measurable set A.<br />

Observe that <strong>the</strong> simple argument reduc<strong>in</strong>g <strong>the</strong> study <strong>of</strong> an X valued measure which<br />

is absolutely cont<strong>in</strong>uous with respect to a f<strong>in</strong>ite control measure to <strong>the</strong> case where <strong>the</strong> X<br />

valued measure satisfies (14) yields ano<strong>the</strong>r characterization <strong>of</strong> <strong>the</strong> RNP; more precisely,<br />

that <strong>the</strong> (separable) Radon-Nikodym <strong>the</strong>orem holds <strong>in</strong> X if and only if for each operator T<br />

from an L 1 (Ix) space with tx f<strong>in</strong>ite (and separable) <strong>in</strong>to X <strong>the</strong>re is an X valued measurable<br />

function g so that Tf = f f g dix for all f <strong>in</strong> L I(IX). From this it is easy to see that if<br />

X is separable and <strong>the</strong> separable Radon-Nikod3)m <strong>the</strong>orem holds <strong>in</strong> X <strong>the</strong>n <strong>the</strong> Radon-<br />

Nikod2~m <strong>the</strong>orem holds <strong>in</strong> X. Indeed, let IX be a f<strong>in</strong>ite measure on a cr-algebra B and let<br />

T'LI(Ix) --+ X be an operator. S<strong>in</strong>ce X is separable <strong>the</strong>re is a sequence {X*}n~ <strong>in</strong> X*<br />

which separates <strong>the</strong> po<strong>in</strong>ts <strong>of</strong> X. Let A be a countably generated sub ~r-algebra <strong>of</strong> 13 so<br />

that all <strong>the</strong> Loc(Ix) functions T* x n * are A-measurable. S<strong>in</strong>ce {Xn}n=l , oc separates po<strong>in</strong>ts <strong>of</strong><br />

X, Tf = TE(f I A) for each f <strong>in</strong> L1 (Ix). The restriction <strong>of</strong> Ix to A is a separable measure<br />

s<strong>in</strong>ce A is countably generated. Thus we get an X valued A-measurable function g so that<br />

Tf = f f. g dIx for each A-measurable function f <strong>in</strong> L l (IX). But <strong>the</strong>n for a general f <strong>in</strong><br />

L1 (Ix) we have Tf = TE(f I A) = f E(f I A)g dIx = f f. g dIx s<strong>in</strong>ce g is A-measurable.<br />

One <strong>of</strong> many places where <strong>the</strong> RNP arises naturally is <strong>in</strong> <strong>the</strong> study <strong>of</strong> vector valued L p<br />

spaces. There is a natural isometric identification <strong>of</strong> Lp,(Ix, X*), 1/p + 1/p* = 1, with<br />

a subspace <strong>of</strong> Lp(Ix, X)*, and for 1 ~ p < (x~, Lp(Ix, X)* = Lp,(Ix, X*) for allf<strong>in</strong>ite (or<br />

cr-f<strong>in</strong>ite) measures Ix if and only if X* has <strong>the</strong> RNP (see [8, Chapter IV]).<br />

There are o<strong>the</strong>r important analytic characterizations <strong>of</strong> spaces with <strong>the</strong> RNP <strong>in</strong> terms <strong>of</strong><br />

mart<strong>in</strong>gales. In particular, <strong>the</strong> RNP spaces are exactly those <strong>Banach</strong> spaces <strong>in</strong> which <strong>the</strong><br />

mart<strong>in</strong>gale convergence <strong>the</strong>orem is valid <strong>in</strong> <strong>the</strong> sense that X has <strong>the</strong> RNP if and only if<br />

every L1 bounded X valued mart<strong>in</strong>gale converges a.e. (see [8, Chapter V]).

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 41<br />

It turns out that <strong>in</strong> many places where one might assume reflexivity <strong>in</strong> order to use weak<br />

compactness <strong>of</strong> <strong>the</strong> unit ball it suffices to assume that <strong>the</strong> space has <strong>the</strong> RNP.<br />

We now discuss <strong>the</strong> differentiability <strong>of</strong> (real valued) convex cont<strong>in</strong>uous functions on a<br />

<strong>Banach</strong> space X. Part <strong>of</strong> <strong>the</strong> importance <strong>of</strong> this topic derives from <strong>the</strong> fact that <strong>the</strong> norm<br />

is a convex cont<strong>in</strong>uous function and differentiability <strong>of</strong> <strong>the</strong> norm is <strong>in</strong>tr<strong>in</strong>sically related to<br />

its smoothness. The most elementary reference for <strong>the</strong> differentiability <strong>of</strong> convex functions<br />

and related topics is probably [11, Chapter 5].<br />

An easy consequence <strong>of</strong> <strong>the</strong> def<strong>in</strong>ition is that a locally bounded convex function is cont<strong>in</strong>uous<br />

and even locally Lipschitz. By us<strong>in</strong>g <strong>the</strong> separation <strong>the</strong>orem <strong>in</strong> X | IR it follows<br />

that whenever f is convex and cont<strong>in</strong>uous <strong>in</strong> a neighborhood <strong>of</strong> a po<strong>in</strong>t x0 <strong>the</strong> set (called<br />

<strong>the</strong> subdifferential <strong>of</strong>f) Of(XO) :"- {X* E X*: x*(x -- xo) ~ f(x) -- f(xo) for all x 6 X} is<br />

nonempty. From <strong>the</strong> <strong>the</strong>ory <strong>of</strong> convex functions on R we know that for each u <strong>the</strong> right and<br />

left derivatives <strong>of</strong> <strong>the</strong> function t ~ f (xo + tu) exist at t = 0. These one-sided derivatives<br />

agree for every u (that is, all directional derivatives exist at x0) if and only if Of(xo) is<br />

a s<strong>in</strong>gle po<strong>in</strong>t which is <strong>the</strong>n necessarily <strong>the</strong> G-derivative <strong>of</strong> f at x0. Consequently f is<br />

G-differentiable at x0 if and only if for every u, f(xo + tu) + f(xo - tu) - 2f(x0) -- o(t)<br />

as t --+ 0.<br />

By consider<strong>in</strong>g f(x) = Ilxll we recover <strong>the</strong> fact mentioned <strong>in</strong> Section 6 that <strong>the</strong> norm<br />

is G-differentiable at x0 <strong>in</strong> <strong>the</strong> unit sphere <strong>of</strong> X if and only if x0 is a smooth po<strong>in</strong>t <strong>of</strong> <strong>the</strong><br />

unit ball <strong>of</strong> X. It also follows that f is F-differentiable at x0 if and only if f(xo + u) +<br />

f (xo - u) - 2f(x0) = o(llull) as Ilull -+ 0. If a convex function f is F-differentiable <strong>in</strong> a<br />

neighborhood <strong>of</strong>xo <strong>the</strong>n D f (x) is cont<strong>in</strong>uous <strong>the</strong>re; that is, F-differentiability <strong>of</strong> a convex<br />

function on an open set implies that it is C 1 <strong>the</strong>re. Indeed, suppose that Xn --+ xo and set<br />

tO* "- Df(xo); u* "-- Df(xn). Given E > 0 <strong>the</strong>re is 6 > 0 so that f(xo + y) - f(xo) -<br />

w*(y) /(~/2)llu~* - w*ll<br />

for all n. Then s<strong>in</strong>ce u, * - Df(xn), we have by <strong>the</strong> convexity <strong>of</strong> f that<br />

6 + tO* (Yn) + f (xo) >~ f (xo + y/7) >~ u* (Yn - Xn + xo) + f (Xn)<br />

or<br />

(~/2)llun* -- tO* II ~< (Un* -- tO* )(Yn)

42 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

The space X is uniformly smooth if <strong>the</strong> norm is uniformly F-differentiable on <strong>the</strong> unit<br />

sphere; that is, if <strong>the</strong> limit limt~0 t -1 (llx + tull - Ilxll) exists uniformly <strong>in</strong> both x and u<br />

on <strong>the</strong> unit sphere.<br />

The classical G~teaux differentiability <strong>the</strong>orem for convex functions says: A cont<strong>in</strong>uous<br />

convex function f on a separable <strong>Banach</strong> space X is G-differentiable on a dense G~<br />

set. Indeed, if {Xn }n--1 ~ is dense <strong>in</strong> X <strong>the</strong>n <strong>the</strong> set <strong>of</strong> G-differentiability <strong>of</strong> f is <strong>the</strong> set<br />

('~n,m Gn,m, where Gn,m is <strong>the</strong> set <strong>of</strong> po<strong>in</strong>ts x <strong>in</strong> X for which <strong>the</strong>re exists 6 > 0 so that<br />

f(x + 6Un) + f(x - gUn) - 2f(x) ~< 6/m. It is readily verified that each Gn,m is open and<br />

dense.<br />

For Fr6chet differentiability, <strong>the</strong> situation is much different even for norms. The norm <strong>of</strong><br />

el is G-differentiable at any po<strong>in</strong>t all <strong>of</strong> whose coord<strong>in</strong>ates are nonzero but is nowhere F-<br />

differentiable. This is typical <strong>in</strong> <strong>the</strong> sense that if X is separable and X* is nonseparable <strong>the</strong>n<br />

X admits an equivalent norm that is nowhere F-differentiable (see [11, Theorem 106]). On<br />

<strong>the</strong> o<strong>the</strong>r hand: If X* is separable <strong>the</strong>n every convex function f on X is F-differentiable<br />

on a dense G~. That <strong>the</strong> set <strong>of</strong> po<strong>in</strong>ts <strong>of</strong> F-differentiability is a G~ is easy to check (and<br />

for this no separability assumption is needed); it is equal to ("In Gn where Gn is <strong>the</strong> set<br />

<strong>of</strong> po<strong>in</strong>ts x <strong>in</strong> X for which <strong>the</strong>re exists 6 > 0 so that suPllull

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 43<br />

8. F<strong>in</strong>ite dimensional <strong>Banach</strong> spaces<br />

S<strong>in</strong>ce all f<strong>in</strong>ite dimensional spaces <strong>of</strong> <strong>the</strong> same dimension over <strong>the</strong> same scalar field are<br />

mutually isomorphic, for results on f<strong>in</strong>ite dimensional spaces to be mean<strong>in</strong>gful <strong>the</strong>y must<br />

be <strong>of</strong> a quantitative nature. The notion <strong>of</strong> <strong>Banach</strong>-Mazur distance is <strong>of</strong> central importance<br />

<strong>in</strong> this context. Evaluat<strong>in</strong>g or even estimat<strong>in</strong>g <strong>the</strong> distance between spaces is <strong>of</strong>ten hard<br />

s<strong>in</strong>ce it generally is quite difficult to f<strong>in</strong>d an operator T for which IIT II II T-1II is m<strong>in</strong>imal<br />

or close to m<strong>in</strong>imal.<br />

We illustrate <strong>the</strong> computation <strong>of</strong> <strong>the</strong> <strong>Banach</strong>-Mazur distance by evaluat<strong>in</strong>g (or <strong>in</strong> some<br />

cases just giv<strong>in</strong>g <strong>the</strong> order <strong>of</strong> magnitude) <strong>the</strong> quantity d (s s While relatively easy, even<br />

<strong>in</strong> this simple situation it is by no means trivial to calculate or even closely estimate <strong>the</strong><br />

distance when p and r are on different sides <strong>of</strong> two. The topic <strong>of</strong> <strong>Banach</strong>-Mazur distances<br />

between f<strong>in</strong>ite dimensional space is treated <strong>in</strong> many more <strong>in</strong>volved situations <strong>in</strong> [20].<br />

Denote <strong>the</strong> formal identity mapp<strong>in</strong>g from ~ to ~n by Ip,r. It is trivial to check that<br />

IIIp,rll = 1 when p ~ r. Consequently, if 1 ~< p < r ~<<br />

ec, <strong>the</strong>n d(gp, ~) ~< n 1/p-1/r. The simplest and most important case occurs when p (or<br />

r) is two. Ei<strong>the</strong>r by <strong>in</strong>duction from <strong>the</strong> case n -- 2, where <strong>the</strong> follow<strong>in</strong>g equality is just <strong>the</strong><br />

parallelogram law, or by us<strong>in</strong>g <strong>the</strong> orthonormality <strong>of</strong> a Rademacher sequence and tak<strong>in</strong>g<br />

<strong>the</strong> vectors xi to be <strong>in</strong>, e.g., L2(0, 1) and exchang<strong>in</strong>g <strong>the</strong> order <strong>of</strong> <strong>in</strong>tegration over [0, 1]<br />

with <strong>the</strong> expectation, one sees that vectors x l ..... x/7 <strong>in</strong> a Hilbert space satisfy <strong>the</strong> identity<br />

~6iXi<br />

i=1<br />

2<br />

- Ilxi ll2.<br />

i--1<br />

(15)<br />

Suppose that 2 < r ~< cx~ and T's --+ ~ is an isomorphism normalized to sat-<br />

isfy IIT-1II = 1 (so that IlTx[]2 /~ []Xl[r for all x). Denot<strong>in</strong>g as usual <strong>the</strong> unit vectors<br />

basis as {ei}, we see from (15) that n

44 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

By consider<strong>in</strong>g Vn* as <strong>the</strong> composition I2,1V*Icc,2 we see that IlVn*ll~,l ~< n. S<strong>in</strong>ce<br />

Vn* = Vn 1 , it follows that d(g.~, gn)

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 45<br />

ot(T) Y<br />

is def<strong>in</strong>ed by lett<strong>in</strong>g Tkei be xi when i --/= k and Tkek = x.<br />

We turn to <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Lewis' lemma. S<strong>in</strong>ce <strong>the</strong> volume <strong>of</strong> T C for any measurable set<br />

is a constant multiple <strong>of</strong> [det(T) Ivol(C), for any operator S:X ---> Y we have<br />

det(To+S)<br />

ot(r0 + S)<br />

~< Idet(T0) I. (18)<br />

Certa<strong>in</strong>ly To must be <strong>in</strong>vertible, so by divid<strong>in</strong>g (18) by Idet(T0)l we can rewrite (18) as<br />

Idet(Ix + To'S) l

46 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

N ~< 2n2; see [20, 8.6].) Hav<strong>in</strong>g done <strong>the</strong>se prelim<strong>in</strong>aries, we can state a nice geometric<br />

reformulation <strong>of</strong> <strong>the</strong> case or(.) = I1" II <strong>in</strong> Lewis' lemma. Aga<strong>in</strong> identify both spaces X and<br />

Y with ~n or C n. When one convex body conta<strong>in</strong>s ano<strong>the</strong>r, a po<strong>in</strong>t x is called a contact<br />

po<strong>in</strong>t <strong>of</strong> <strong>the</strong> bodies if it is <strong>in</strong> <strong>the</strong> <strong>in</strong>tersection <strong>of</strong> <strong>the</strong>ir boundaries (so if <strong>the</strong> bodies are unit<br />

balls associated with two norms, a contact po<strong>in</strong>t is a po<strong>in</strong>t <strong>in</strong> <strong>the</strong> <strong>in</strong>tersection <strong>of</strong> <strong>the</strong> two<br />

unit spheres).<br />

Assume that Bx C By and vol(Bx) ~> vol(TBx) for every operator T on ~n [C n] for<br />

which TBx C Br. Then <strong>the</strong>re exist contactpo<strong>in</strong>ts xl ..... XN <strong>of</strong> Bx and Br and contact<br />

po<strong>in</strong>ts x 1 .... , x N <strong>of</strong> Bx, and Br', and ck ~ 0 so that I = ~=l ckx~ | xk. Also, N

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 47<br />

each n spaces Xn and Yn <strong>of</strong> dimension n so that <strong>in</strong>fd(Xn, Yn)/n > 0. If one puts 2n pairs<br />

<strong>of</strong> symmetric po<strong>in</strong>ts on <strong>the</strong> unit sphere <strong>of</strong> ~, where <strong>the</strong> po<strong>in</strong>ts are chosen <strong>in</strong>dependently<br />

and are distributed uniformly on that sphere, and <strong>the</strong>n takes <strong>the</strong> symmetric convex hull <strong>of</strong><br />

<strong>the</strong> union <strong>of</strong> <strong>the</strong>se po<strong>in</strong>ts with <strong>the</strong> unit vector basis, one obta<strong>in</strong>s a unit ball for a (random)<br />

space. If one takes two <strong>of</strong> <strong>the</strong>se random spaces, <strong>the</strong>n with big probability (that is, with<br />

probability tend<strong>in</strong>g to one as n --+ cx~) <strong>the</strong> <strong>Banach</strong>-Mazur distance between <strong>the</strong> two spaces<br />

exceeds gn for some constant g > 0 <strong>in</strong>dependent <strong>of</strong> n. Although this construction is easy to<br />

describe, <strong>the</strong> computations are delicate and <strong>the</strong> reader is referred to [20, 38.1 ] for details.<br />

The probabilistic approach has many o<strong>the</strong>r applications. For example, from <strong>the</strong> estimate<br />

d(X, s 0 <strong>the</strong>re exists no - no(e, k) so that if dim(X) ~ no <strong>the</strong>n X conta<strong>in</strong>s a<br />

subspace Y with d(Y, ~) < 1 + e. There are known good estimates on n0(e, k) (and even<br />

better estimates for special classes <strong>of</strong> spaces). For general spaces n0(e, k) ~< exp(otk/e 2)<br />

for some constant a. Dvoretzky's <strong>the</strong>orem is treated <strong>in</strong> detail <strong>in</strong> several books, <strong>in</strong>clud<strong>in</strong>g<br />

[9, 19.1], [17, 4.3], [16, 1.5.8]. An exposition <strong>of</strong> Dvoretzky's <strong>the</strong>orem and related results<br />

is given <strong>in</strong> [28]. There are many pro<strong>of</strong>s <strong>of</strong> Dvoretzky's <strong>the</strong>orem and <strong>in</strong> most <strong>of</strong> <strong>the</strong>m <strong>the</strong><br />

Dvoretzky-Rogers lemma is <strong>the</strong> first step.<br />

From Dvoretzky's <strong>the</strong>orem and <strong>the</strong> technique for construct<strong>in</strong>g basic sequences discussed<br />

<strong>in</strong> Section 3 one gets <strong>in</strong> any <strong>in</strong>f<strong>in</strong>ite dimensional <strong>Banach</strong> space a basic sequence {Xn}n~=l<br />

.2k+l<br />

so that for each k, {xj }j__Zk§ 1 is 1/k-equivalent to an orthonormal basis <strong>in</strong> a Hilbert space.

48 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

An easy consequence <strong>of</strong> this is that for every square summable sequence {C~n}n~__l, <strong>of</strong><br />

scalars, every <strong>in</strong>f<strong>in</strong>ite dimensional <strong>Banach</strong> space conta<strong>in</strong>s an unconditionally convergent<br />

series ~n Yn such that for each n, IlYn II = I~nl. One can also easily deduce this from <strong>the</strong><br />

simple Dvoretzky-Rogers lemma; <strong>in</strong> fact, historically it was this application that motivated<br />

<strong>the</strong> discovery <strong>of</strong> <strong>the</strong> Dvoretzky-Rogers lemma.<br />

A result related to Dvoretzky's <strong>the</strong>orem is Kriv<strong>in</strong>e's <strong>the</strong>orem:<br />

If {Xn }n~__l is a basic sequence <strong>in</strong> a <strong>Banach</strong> space, <strong>the</strong>n <strong>the</strong>re exists 1 0 <strong>the</strong>re is a<br />

lattice isomorphism T from g n p <strong>in</strong>to X with II T II II T-1 II < 1 + ~.<br />

From Kriv<strong>in</strong>e's <strong>the</strong>orem and <strong>the</strong> fact discussed <strong>in</strong> Section 4 that ~2 embeds isometrically<br />

<strong>in</strong>to L p (0, 1) it is easy to deduce Dvoretzky's <strong>the</strong>orem. This is not <strong>the</strong> recommended<br />

route to Dvoretzky's <strong>the</strong>orem as it is difficult to navigate through Kriv<strong>in</strong>e's <strong>the</strong>orem. More<br />

importantly, <strong>the</strong> tight quantitative estimates obta<strong>in</strong>able from direct pro<strong>of</strong>s <strong>of</strong> Dvoretzky's<br />

<strong>the</strong>orem have many applications.<br />

If we apply Kriv<strong>in</strong>e's <strong>the</strong>orem to a basic sequence which is equivalent to <strong>the</strong> unit vector<br />

basis for ~r we see immediately that <strong>the</strong> only p that is obta<strong>in</strong>able is p = r. Thus by apply<strong>in</strong>g<br />

Kriv<strong>in</strong>e's <strong>the</strong>orem to a disjo<strong>in</strong>t sequence <strong>in</strong> an <strong>in</strong>f<strong>in</strong>ite dimensional L p (#) space we <strong>in</strong>fer<br />

that any equivalent renorm<strong>in</strong>g <strong>of</strong> Lp(#) conta<strong>in</strong>s for every k and ~ > 0 a subspace whose<br />

<strong>Banach</strong>-Mazur distance to g k p is less than 1 + E This implies that an <strong>in</strong>f<strong>in</strong>ite dimensional<br />

Lp(#) space cannot be given an equivalent norm which has a better modulus <strong>of</strong> convexity<br />

or smoothness than its natural norm.<br />

Two notions that are very important for both <strong>the</strong> f<strong>in</strong>ite dimensional and <strong>in</strong>f<strong>in</strong>ite dimensional<br />

<strong>the</strong>ories are that <strong>of</strong> type and cotype. A <strong>Banach</strong> space X is said to have type p provided<br />

<strong>the</strong>re is a constant C so that for every sequence x l ..... xn <strong>in</strong> X,<br />

E • 6iXi<br />

i=1<br />

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 49<br />

a constant C so that for every sequence X l ..... Xn <strong>in</strong> X,<br />

[[xill q<br />

i=1<br />

2) 1/2<br />

The best constants <strong>in</strong> (22) and (23) are denoted by Tp (X) and Cq (X). By <strong>the</strong> parallelogram<br />

identity characterization <strong>of</strong> Hilbert space, a space X is isometric to a Hilbert space if and<br />

only if Tz(X) = 1 = C2(X). Even a one dimensional space does not have type p for any<br />

p > 2 or cotype q for any q < 2. For every space X, 7"1 (X) = 1 and Coo(X) -- 1 by<br />

convexity <strong>of</strong> <strong>the</strong> norm. As functions <strong>of</strong> p and q, Tp(X) is nondecreas<strong>in</strong>g and Cp(X) is<br />

non<strong>in</strong>creas<strong>in</strong>g. The <strong>in</strong>equalities Tp(X)

50 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

sional L~(#) spaces are universal for separable spaces, <strong>the</strong>y do not have type p for any<br />

p > 1. Similarly, an <strong>in</strong>f<strong>in</strong>ite dimensional Lq (#) space for 2 ~< q ~< oo does not have cotype<br />

smaller than q.<br />

If {Xn }n=l ~ is a sem<strong>in</strong>ormalized unconditionally basic sequence <strong>in</strong> Lq(lZ), 2 < q < oo,<br />

with # a probability measure and <strong>in</strong>f IlXn 112 > 0, <strong>the</strong>n s<strong>in</strong>ce Lq(#) has type 2 and L2(#) has<br />

cotype 2 and I1" 112 ~ I1" IIq we <strong>in</strong>fer that {Xn}n~=l is, <strong>in</strong> Lq(#), equivalent to <strong>the</strong> unit vector<br />

basis <strong>of</strong> g2. Thus such a sequence cannot have dense l<strong>in</strong>ear span <strong>in</strong> Lq(#). In particular,<br />

<strong>the</strong> trigonometric system is not unconditional <strong>in</strong> Lq (0, 1) for 2 < q ~< c~ and, by duality,<br />

also not <strong>in</strong> L p(O, 1), 1

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 51<br />

One can assume that I[x + YI[ + [Ix - YI[ - 2 with t "- ]]x+yll-llx-y[] ~ O. Then for e -- +1,<br />

2<br />

1 ~Crp 1 - Crp<br />

Ilx + opey II ~< IIx + eyl[ + 2 I[x - eyl[- 1 + eCrpt.<br />

Therefore<br />

(EII + + ,/1 + t2- (EII + lyl12)<br />

F<strong>in</strong>ally, iterate (27) to obta<strong>in</strong> (25). Formally, for k -- 1 ..... n def<strong>in</strong>e Sk "-- ~ki= 1 6iXi; we<br />

need to show that (EIIx + o'pSkllP) 1/p

52 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

arose first <strong>in</strong> <strong>the</strong> study <strong>of</strong> probability <strong>in</strong> <strong>Banach</strong> spaces and are sometimes called B-convex<br />

(see [34]).<br />

Similarly, a space does not have f<strong>in</strong>ite cotype if and only if it conta<strong>in</strong>s for every n arbi-<br />

trarily close copies <strong>of</strong> s S<strong>in</strong>ce ek 1 is a subspace <strong>of</strong> ~ (isometrically <strong>in</strong> <strong>the</strong> case <strong>of</strong> real<br />

scalars and up to constant <strong>in</strong> <strong>the</strong> complex case), we deduce that if X is B-convex <strong>the</strong>n it<br />

has cotype q for some q < cx~. S<strong>in</strong>ce a uniformly convex space cannot have subspaces <strong>of</strong><br />

arbitrary dimension which are arbitrarily close to s it follows that a uniformly convex<br />

space is B-convex; that is, has nontrivial type and cotype.<br />

The relation <strong>of</strong> B-convexity to reflexivity is not so simple but has been clarified. A B-<br />

convex space need not be reflexive-<strong>the</strong>re is even a nonreflexive space <strong>of</strong> type 2. Though<br />

nonreflexive <strong>Banach</strong> spaces need not conta<strong>in</strong> almost isometric copies <strong>of</strong> s <strong>the</strong>y do conta<strong>in</strong><br />

configurations <strong>of</strong> <strong>the</strong> follow<strong>in</strong>g type for every n and ~ > 0: norm one vectors x i .... , Xn<br />

so that for every k < n, Ilxl +"-+ Xk - (Xk+l + '''-~- xn)ll >~ n - E. In particular, tak<strong>in</strong>g<br />

n : 2 we see that: Every nonreflexive space conta<strong>in</strong>s real subspaces arbitrarily close<br />

to real s That is, a real space whose unit ball does not have a two dimensional section<br />

arbitrarily close to a square must be reflexive. See [2, 4.111] for pro<strong>of</strong>s <strong>of</strong> <strong>the</strong>se results.<br />

We next discuss <strong>the</strong> duality <strong>the</strong>ory <strong>of</strong> type and cotype. It is simple that if X has type p<br />

<strong>the</strong>n X* has cotype p* and Cp,(X*)

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 53<br />

would be bounded, where P is <strong>the</strong> orthogonal projection onto <strong>the</strong> closed l<strong>in</strong>ear span <strong>of</strong> <strong>the</strong><br />

Rademacher sequence {en}~_l (a space X for which P is bounded is said to be K-convex).<br />

/5 is def<strong>in</strong>ed explicitly for f/ <strong>in</strong> L2(#) and xi <strong>in</strong> X by ['(~i~=l fixi) -- Y~"i=l (P~)xi" It<br />

is a deep fact (see [16, II.14], [17, 2.4], and [38]) that: X is K-convex if and only if X is<br />

B-convex. Thus <strong>the</strong> presence <strong>of</strong> copies <strong>of</strong> ~ is exactly <strong>the</strong> factor which h<strong>in</strong>ders a clean<br />

duality between type and cotype. If X is B-convex and <strong>of</strong> cotype p* <strong>the</strong>n X* is <strong>of</strong> type p.<br />

For f<strong>in</strong>ite dimensional X it is important to estimate <strong>the</strong> norm <strong>of</strong> <strong>the</strong> Rademacher projection<br />

(called <strong>the</strong> K-convexity constant <strong>of</strong> X and usually denoted by K(X)). A useful<br />

estimate (which is valid also for isomorphs <strong>of</strong> <strong>in</strong>f<strong>in</strong>ite dimensional Hilbert space) is<br />

K(X) 1, <strong>the</strong> distance estimate from John's <strong>the</strong>orem<br />

gives that K (X) ~< K l log n. These estimates are sharp up to <strong>the</strong> values <strong>of</strong> <strong>the</strong> constants K<br />

and Kj. For a discussion see [20, pp. 86-92], [17], [16, II.14], or [38].<br />

9. Local structure <strong>of</strong> <strong>in</strong>f<strong>in</strong>ite dimensional spaces<br />

In this section we describe results and techniques whose purpose is to relate <strong>the</strong> structure <strong>of</strong><br />

an <strong>in</strong>f<strong>in</strong>ite dimensional <strong>Banach</strong> space with <strong>the</strong> structure <strong>of</strong> its f<strong>in</strong>ite dimensional subspaces.<br />

We will be particularly <strong>in</strong>terested <strong>in</strong> properties <strong>of</strong> a space which depend only on its family<br />

<strong>of</strong> f<strong>in</strong>ite dimensional subspaces and not on <strong>the</strong> way <strong>the</strong>se f<strong>in</strong>ite dimensional spaces are<br />

"glued toge<strong>the</strong>r" to form <strong>the</strong> <strong>in</strong>f<strong>in</strong>ite dimensional space.<br />

A <strong>Banach</strong> space X is said to be L-representable <strong>in</strong> a <strong>Banach</strong> space Y if for every f<strong>in</strong>ite<br />

dimensional subspace E <strong>of</strong> X and every ~ > 0 <strong>the</strong>re is a f<strong>in</strong>ite dimensional subspace F <strong>of</strong><br />

Y with d(E, F)

54 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

so that if (1 + 3)-l ~< II Sy II ~ 1 + 6 for every y <strong>in</strong> a 6-net <strong>in</strong> <strong>the</strong> unit sphere <strong>of</strong> Y, and some<br />

operator S, <strong>the</strong>n (1 + ~)-l [ly II ~< IlSy II ~< (1 + ~)IlY II for all y E Y.<br />

We pass to <strong>the</strong> pro<strong>of</strong> <strong>of</strong> <strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong> local reflexivity.<br />

Let e > 0 and let 6 = 6(e) be as above. Let E and F be f<strong>in</strong>ite dimensional subspaces<br />

<strong>of</strong> X**, respectively, X*. We pick {ijj}j__ 1, m <strong>in</strong> <strong>the</strong> unit ball <strong>of</strong> X* so that <strong>the</strong> set conta<strong>in</strong>s<br />

an algebraic basis <strong>of</strong> F and so that IIx**ll ~< (1 + 6)maxj Ix**(v~)l for every x** E E.<br />

Let {wi**}/n__l be a 6-net <strong>in</strong> <strong>the</strong> unit sphere <strong>of</strong> E so that {w**}~ 1 is a basis <strong>of</strong> E N X and<br />

{//)i**}i=1 r is a basis <strong>of</strong> E for some k ~< r < n<br />

"<br />

Write w i **<br />

=Zh=l)~ihW<br />

r h ** for r < i ~ n.<br />

For r < i ~< n put #i,h -- Xi,h if h ~ r, ~ii = -- 1, and #i,h = 0 for r < h # i. Consider <strong>the</strong><br />

operator<br />

/7<br />

n-r+k<br />

s. (2 e . . . 9 x)~ --, (x ~ i . . 9 x)~<br />

d~<br />

def<strong>in</strong>ed by<br />

( )<br />

S(xl ..... Xn)-- x1 ..... Xk, #r+l,hXh ..... #n,hXh 9<br />

h-1 h-1<br />

By <strong>the</strong> choice <strong>of</strong> #i,h for h > r it follows that S is onto. Hence <strong>the</strong> operator<br />

/7<br />

n-r+h<br />

~. (x e . . . 9 x)~ --, (x e . . . 9 :~ 9 R'm)~<br />

def<strong>in</strong>ed by S(xl ..... Xn) -- (S(xl ..... Xn), vj(xh)), 1 ~ j

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 55<br />

Recall that a family b/<strong>of</strong> subsets <strong>of</strong> a set I is called a filter if it is closed under f<strong>in</strong>ite<br />

<strong>in</strong>tersections, does not conta<strong>in</strong> <strong>the</strong> empty set, and whenever A C B with A E b/<strong>the</strong>n B<br />

b/. A maximal (with respect to <strong>in</strong>clusion) filter is called an ultrafilter. By Zorn's lemma<br />

every filter is conta<strong>in</strong>ed <strong>in</strong> an ultrafilter. An ultrafilter is called free (or nontrivial) if <strong>the</strong><br />

<strong>in</strong>tersection <strong>of</strong> all sets <strong>in</strong> b/is empty. An <strong>in</strong>dexed family {Xi}i6I <strong>in</strong> a topological space<br />

is said to converge to x with respect to a filter bt (<strong>in</strong> symbols, x -- l<strong>in</strong>~ xi ) provided for<br />

every open set G conta<strong>in</strong><strong>in</strong>g x <strong>the</strong> set {i: xi ~ G} belongs to/g. A Hausdorff space is<br />

compact if and only if every <strong>in</strong>dexed family {xi }i~I converges (to a unique po<strong>in</strong>t) for every<br />

free ultrafilter b/on I. Assume now that I is a set and b/is a free ultrafilter on I; assume<br />

also that for all i, Xi is a <strong>Banach</strong> space. We def<strong>in</strong>e a sem<strong>in</strong>orm Ill'Ill on (Y-~i Xi)ec by<br />

IIIxlll = lirr~ Ilxi II where x -- {xi }i~/with xi ~ Xi for all i. The limit exists s<strong>in</strong>ce a closed<br />

bounded <strong>in</strong>terval on <strong>the</strong> l<strong>in</strong>e is compact. The quotient <strong>of</strong> (~-~i Xi)ec with respect to <strong>the</strong><br />

closed subspace <strong>of</strong> all x with ]llx Ill --- 0 with its obvious norm is a <strong>Banach</strong> space, called <strong>the</strong><br />

ultraproduct <strong>of</strong> <strong>the</strong> Xi (with respect to b/), and is denoted by (1--Ii Xi)cr If all <strong>the</strong> Xi are<br />

<strong>the</strong> same space X we call <strong>the</strong> space thus obta<strong>in</strong>ed an ultrapower <strong>of</strong> X, denoted also by Xu.<br />

Ultraproducts <strong>of</strong> <strong>Banach</strong> spaces are treated <strong>in</strong> detail <strong>in</strong> [9, Chapter 8].<br />

Given two families {Xi}icI and {Yi}i~I <strong>of</strong> spaces and operators 7):Xi ~ Yi with<br />

sup/ ]]Ti ]] < cx:), <strong>the</strong>re is a natural operator T:(1-Ii Xi)lg -'+ (Hi Yi)bt called <strong>the</strong> ultraproduct<br />

<strong>of</strong><strong>the</strong> operators 7). It maps an element <strong>in</strong> (1-Ii xi)cr represented by x = {xi}i6I <strong>in</strong><br />

(~_~ Xi)~ <strong>in</strong>to <strong>the</strong> element <strong>in</strong> (]-Ii Yi)Cr represented by y = {~xi}is/.<br />

The ultraproduct <strong>of</strong> one dimensional spaces is one dimensional and more generally if<br />

dim Xi = n < ec for all i <strong>the</strong>n (I-Ii xi)cr is also n-dimensional. On <strong>the</strong> o<strong>the</strong>r hand if I = N<br />

and limu (dim Xi) = ec <strong>the</strong>n (I-Ii xi)u is already nonseparable.<br />

The ultraproduct <strong>of</strong> <strong>Banach</strong> lattices is aga<strong>in</strong> a <strong>Banach</strong> lattice if we take as <strong>the</strong> positive<br />

cone <strong>in</strong> (Hi xi)cr <strong>the</strong> set <strong>of</strong> all elements which have representatives x = {xi}iEl <strong>in</strong><br />

(Y]~i Xi)oc with xi ~ 0 for all i. If all <strong>the</strong> Xi are abstract Lp spaces for some fixed p,<br />

1 ~< p < ec, <strong>the</strong>n (Hi xi)u is aga<strong>in</strong> an abstract Lp space and hence is isometric to Lp(#)<br />

for some measure # by <strong>the</strong> L p version <strong>of</strong> <strong>the</strong> Kakutani representation <strong>the</strong>orem. Similarly,<br />

if all <strong>the</strong> Xi are C (Ki) spaces for some compact Hausdorff Ki <strong>the</strong>n so is (I-Ii xi)cr However,<br />

for o<strong>the</strong>r families <strong>of</strong> <strong>Banach</strong> spaces Xi (even, e.g., if all are Orlicz spaces with <strong>the</strong><br />

same Orlicz function q)) <strong>the</strong> determ<strong>in</strong>ation <strong>of</strong> <strong>the</strong> nature <strong>of</strong> (1-[i Xi)u is not an easy task.<br />

As a first application <strong>of</strong> ultraproducts we shall prove now a fact mentioned already <strong>in</strong><br />

Section 4: If 1

56 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

(and <strong>in</strong> particular a reflexive space) <strong>the</strong>re is a contractive projection from Xu onto <strong>the</strong><br />

canonical image <strong>of</strong> X <strong>in</strong> it. Map {Xi}iEI to w*-l<strong>in</strong>gr xi. As for duality, <strong>the</strong> space (I-Ii x*)u<br />

can be <strong>in</strong> a natural way identified with a subspace (usually proper) <strong>of</strong> <strong>the</strong> dual <strong>of</strong> (1--Ii xi)ct.<br />

We now discuss <strong>the</strong> relation between ultraproducts and )~-representability. It follows<br />

directly from <strong>the</strong> def<strong>in</strong>itions that any ultrapower Xct <strong>of</strong> a <strong>Banach</strong> space X is f<strong>in</strong>itely representable<br />

<strong>in</strong> X. There is also a converse, <strong>in</strong> a sense, to this statement. A <strong>Banach</strong> space<br />

X is ,k-representable <strong>in</strong> Y if and only if <strong>the</strong>re is a subspace Z <strong>of</strong> some ultraproduct <strong>of</strong> Y<br />

so that d(X, Z) 0. Introduce a partial order on I by<br />

(El, 61) < (E2, 62) if E1 C E2 and 61 > 62, and let///be an ultrafilter on I conta<strong>in</strong><strong>in</strong>g for<br />

all i E I <strong>the</strong> set {j E I: i < j}. By assumption <strong>the</strong>re is for every (E, 6) 6 1 an operator<br />

TE,~ from E <strong>in</strong>to Y so that IIx II ~< II TE,~x II ~< ()~ + 6)Ilxll for all x E E. For every x 6 X<br />

let~ = {xE,~} 6 (Y~'~i Y)~ be def<strong>in</strong>ed by xE,~ = TE,~x ifx 6 E and xE,~ =0ifx ~ E. The<br />

image <strong>of</strong> all <strong>the</strong>se s <strong>in</strong> YU is easily seen to be a subspace Z which satisfies d (X, Z) ~< )~.<br />

From <strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong> local reflexivity it follows that for every <strong>Banach</strong> space X <strong>the</strong><br />

space X** is isometric to a norm one complemented subspace <strong>of</strong> a suitable ultraproduct<br />

<strong>of</strong> X. Without <strong>the</strong> complementation assertion this is a special case <strong>of</strong> <strong>the</strong> result above.<br />

To get <strong>the</strong> complementation assertion we have to modify <strong>the</strong> pro<strong>of</strong> above. Now let I be<br />

<strong>the</strong> set <strong>of</strong> triples (E, F, 6), where E is a f<strong>in</strong>ite-dimensional subspace <strong>of</strong> X**, F a f<strong>in</strong>itedimensional<br />

subspace <strong>of</strong> X*, and 6 > 0. Introduce a partial order on I by (El, F1,61) <<br />

(E2, F2, 62) if El C E2, F1 C F2, and 61 > 62, and let/g be an ultrafilter on I which ref<strong>in</strong>es<br />

<strong>the</strong> partial order filter. For every (E, F, 6) E I let TE,F,e :E ~ X be <strong>the</strong> operator given by<br />

<strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong> local reflexivity. Us<strong>in</strong>g <strong>the</strong>se operators we def<strong>in</strong>e as above an isometry T<br />

from X** <strong>in</strong>to XU. Def<strong>in</strong>e a map S from Xu <strong>in</strong>to X** by S({xi }) = w*-limu xi. From <strong>the</strong><br />

properties <strong>of</strong> {TE,F,~} one deduces easily that ST is <strong>the</strong> identity on X**, so that T S is a<br />

projection <strong>of</strong> norm 1 from XU onto T X**.<br />

A property (P) <strong>of</strong> <strong>Banach</strong> spaces is called a super property provided that if X satisfies<br />

(P) and Y is f<strong>in</strong>itely representable <strong>in</strong> X, <strong>the</strong>n Y satisfies (P). In particular, a super property<br />

passes from a space X to all closed subspaces <strong>of</strong> its ultraproducts. So if (P) is a hereditary<br />

property (i.e., passes to closed subspaces), a <strong>Banach</strong> space X has super (P) if and only if<br />

every ultrapower <strong>of</strong> X has (P). For example, X is superreflexive if every ultrapower <strong>of</strong> X is<br />

reflexive. An explicit local property which characterizes superreflexivity is <strong>the</strong> follow<strong>in</strong>g:<br />

A <strong>Banach</strong> space X is superreflexive if and only if for every 6 > 0 <strong>the</strong>re is an <strong>in</strong>teger N(6) so<br />

that any 6-separated dyadic tree <strong>in</strong> <strong>the</strong> unit ball <strong>of</strong> X has height ~< N (6). By an 6-separated<br />

dyadic tree <strong>of</strong> height N we mean a set <strong>of</strong> po<strong>in</strong>ts {Xi,n: 1

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 57<br />

<strong>in</strong> <strong>the</strong> unit ball <strong>of</strong> X. In a similar manner we get <strong>in</strong> every nonreflexive space a 1-separated<br />

dyadic tree <strong>of</strong> an arbitrary f<strong>in</strong>ite height <strong>in</strong> <strong>the</strong> unit ball.<br />

It is not hard to show that <strong>the</strong> existence <strong>of</strong> arbitrarily tall e-separated trees <strong>in</strong> <strong>the</strong> unit ball<br />

(for some e > 0 or for e = 1) is a selfdual property and thus so is superreflexivity. Much<br />

deeper is <strong>the</strong> fact that a space is superreflexive if and only if it has an equivalent uniformly<br />

convex norm. S<strong>in</strong>ce uniform convexity is def<strong>in</strong>ed by an <strong>in</strong>equality <strong>in</strong>volv<strong>in</strong>g four vectors<br />

<strong>the</strong> if part is obvious. The hard part is <strong>the</strong> only if part; that is, <strong>the</strong> construction <strong>of</strong> a uniformly<br />

convex equivalent norm <strong>in</strong> a space which does not have e-separated dyadic trees <strong>of</strong> large<br />

height <strong>in</strong> its unit ball. The notion <strong>of</strong> dyadic trees rem<strong>in</strong>ds one <strong>of</strong> mart<strong>in</strong>gales and <strong>in</strong> fact <strong>the</strong><br />

most elegant way to prove <strong>the</strong> only if part is to use vector-valued mart<strong>in</strong>gales. The pro<strong>of</strong><br />

shows <strong>in</strong> particular that if X is uniformly convex <strong>the</strong>n <strong>the</strong>re is an equivalent norm whose<br />

modulus <strong>of</strong> convexity ~(~) satisfies ~(e) ~> C~ q for some C > 0 and q < oe. The pro<strong>of</strong> <strong>of</strong><br />

this fact and related material can be found <strong>in</strong> [2, 4.IV] and [6, IV.4] (see also [29]).<br />

We <strong>in</strong>troduce next a class <strong>of</strong> <strong>Banach</strong> spaces def<strong>in</strong>ed <strong>in</strong> terms <strong>of</strong> <strong>the</strong>ir f<strong>in</strong>ite dimensional<br />

subspaces which are closely related to Lp(#) spaces. Let 1 ~< p ~< oe and )~ ~> 1. A <strong>Banach</strong><br />

space X is called an 12p,z space if for every f<strong>in</strong>ite dimensional subspace E <strong>of</strong> X <strong>the</strong>re is<br />

/7<br />

a fur<strong>the</strong>r subspace F D E with d (F, gp) ~< )~ where n -- dim F. A space is called an s<br />

space if it is an s space for some )~ < oo.<br />

It is evident that every Lp(#) space, 1 ~< p ~< oe, is an s space for every e > 0<br />

(take as F a small perturbation <strong>of</strong> a subspace spanned by a f<strong>in</strong>ite number <strong>of</strong> suitable disjo<strong>in</strong>t<br />

<strong>in</strong>dicator functions). Similarly, every C(K) space is an E~,l+~ space for every e > 0 (use<br />

partitions <strong>of</strong> unity). By <strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong> local reflexivity, if X** is an s space <strong>the</strong>n X<br />

is an s space for every ~ > 0. From ultraproduct arguments used already above, it<br />

follows that: Any s space is isomorphic to a subspace <strong>of</strong> Lp(lZ) for some measure #<br />

(1 ~< p ~< ~). In particular, for 1 < p < oe every s space is reflexive and any/22 space<br />

is isomorphic to a Hilbert space.<br />

It is possible to make a stronger statement. Assume first that 1 < p < oe and that X<br />

is an s space. There is thus a set I <strong>of</strong> f<strong>in</strong>ite dimensional subspaces E <strong>of</strong> X, directed<br />

by <strong>in</strong>clusion, with X -- ~EcI E, so that for every E <strong>the</strong>re are operators TE'E --> gp(E),<br />

SE'g~ (E) ---> E so that IITEII ~< 1, IISEII ~< ~ and SETE -- Ie (n(E) --dimE). LetL/be<br />

an ultrafilter on I which ref<strong>in</strong>es <strong>the</strong> order filter and put Y -- (FI gp(E))z4- Def<strong>in</strong>e T from<br />

X to Y by mapp<strong>in</strong>g x to <strong>the</strong> class represented by {TEx}Ecl (<strong>in</strong> view <strong>of</strong> <strong>the</strong> choice <strong>of</strong>/2<br />

it does not matter that TEx is def<strong>in</strong>ed only for E which conta<strong>in</strong> x). Def<strong>in</strong>e S: Y -+ X by<br />

S{yE} = w-lim~ SEyE (recall that X is reflexive). Then ST = Ix and Y is an Lp(#) space.<br />

Consequently: Every E,p space X, 1 < p < oo, is isomorphic to a complemented subspace<br />

<strong>of</strong>an Lp(#) space. If X is separable we deduce that X is isomorphic to a complemented<br />

subspace <strong>of</strong> L p (0, 1). Similar considerations (start<strong>in</strong>g with <strong>the</strong> fact proved <strong>in</strong> Section 4 that<br />

g2 is isometric to a complemented subspace <strong>of</strong> Lp(O, 1)) yield that every Hilbert space is<br />

isometric to a complemented subspace <strong>of</strong> some L p (#) space when 1 < p < oe.<br />

In <strong>the</strong> cases p -- 1 and p -- cx~ we can reason similarly, but now def<strong>in</strong>e S'Y --+ X**<br />

by S{yE} -- w*-limu SEyE and get that ST -- Jx, <strong>the</strong> natural <strong>in</strong>clusion <strong>of</strong> X <strong>in</strong>to X**.<br />

By consider<strong>in</strong>g T**" X** ----> Y** and S**" Y** ----> X (iv) and recall<strong>in</strong>g that <strong>the</strong>re is a norm<br />

one projection from X (iv) onto X** (see Section 2), it follows that if X is an 121 space<br />

<strong>the</strong>n X** is isomorphic to a complemented subspace <strong>of</strong> an L l(#) space. S<strong>in</strong>ce L1 (#)* is

58 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

<strong>in</strong>jective and s<strong>in</strong>ce <strong>the</strong>re is a norm one projection from X*** onto X* we deduce that if X<br />

is an E1 space <strong>the</strong>n X* is an <strong>in</strong>jective <strong>Banach</strong> space. Similarly, if X is an s space <strong>the</strong>n<br />

<strong>the</strong> constructed ultraproduct Y is a C(K) space and X** is isomorphic to a complemented<br />

subspace <strong>of</strong> <strong>the</strong> <strong>in</strong>jective space Y**. That is, if X is an 12~ space <strong>the</strong>n X** is an <strong>in</strong>jective<br />

<strong>Banach</strong> space.<br />

The converse <strong>of</strong> <strong>the</strong> previous statements essentially hold. First we show: Let 1 < p < ~.<br />

A <strong>Banach</strong> space X is isomorphic to a complemented subspace <strong>of</strong> an Lp(#) space if and<br />

only if X is an s space or X is isomorphic to a Hilbert space. To see <strong>the</strong> "only if"<br />

direction, assume that <strong>the</strong>re is a projection Q from Y = Lp(#) onto a subspace X which is<br />

not isomorphic to a Hilbert space. By <strong>the</strong> dichotomy pr<strong>in</strong>ciple for L p spaces, 2 < p < cx~,<br />

discussed <strong>in</strong> Section 4 (and, by duality, also for 1 < p < 2), X has a subspace Z isomorphic<br />

to e p onto which <strong>the</strong>re is a projection, say R. Let E be a f<strong>in</strong>ite dimensional subspace <strong>of</strong> X<br />

and ~ > 0. There is a f<strong>in</strong>ite dimensional subspace F <strong>of</strong> Y (a small perturbation <strong>of</strong> <strong>the</strong> span<br />

<strong>of</strong> disjo<strong>in</strong>t <strong>in</strong>dicator functions) conta<strong>in</strong><strong>in</strong>g E so that d(F, ep)

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 59<br />

<strong>the</strong> structure <strong>of</strong> <strong>the</strong>ir f<strong>in</strong>ite dimensional subspaces and by pass<strong>in</strong>g to <strong>the</strong> duals we get direct<br />

<strong>in</strong>formation only on <strong>the</strong> f<strong>in</strong>ite dimensional quotient spaces.<br />

The s spaces give a nice local description <strong>of</strong> <strong>the</strong> complemented subspaces <strong>of</strong> Lp(O, 1)<br />

for 1 < p < cx~. From <strong>the</strong> global po<strong>in</strong>t <strong>of</strong> view <strong>the</strong>se complemented subspaces are very hard<br />

to describe. Some global structure <strong>the</strong>orems are known (e.g., every separable s space<br />

1 ~< p ~< ~ has a Schauder basis, and every separable <strong>in</strong>f<strong>in</strong>ite dimensional s space 1 <<br />

p < cx~ which does not conta<strong>in</strong> a copy <strong>of</strong> e2 is isomorphic to e p) but many natural questions<br />

on <strong>the</strong>m are still open. For example, it is unknown whe<strong>the</strong>r every separable s space,<br />

1 < p < cx~ (p :/: 2), has an unconditional basis. There are uncountably many dist<strong>in</strong>ct<br />

isomorphism types among <strong>the</strong> separable s spaces, 1 ~< p ~< ~ (p r 2). The simplest<br />

examples for 1 < p 7~ 2 < oc (besides Lp(#) spaces) are ep | e2 and (e2 (~ e2 (~'" ")p.<br />

A discussion <strong>of</strong> all <strong>of</strong> <strong>the</strong>se questions is conta<strong>in</strong>ed <strong>in</strong> [22].<br />

Among <strong>the</strong> separable/2~ spaces <strong>the</strong>re are <strong>the</strong> C(K) spaces with K countable compact<br />

metric which, as was po<strong>in</strong>ted out <strong>in</strong> Section 4, form uncountably many dist<strong>in</strong>ct isomorphism<br />

types. There are however spaces X such that X* is isometric to el but X is not<br />

isomorphic to a C(K) space. Such a space X is an s space for every e > 0. In fact,<br />

a space X is s l+~ for every E > 0 if and only if X* is isometric to LI(#) for some<br />

measure #. It is known that such an X has a subspace isometric to co. In contradist<strong>in</strong>ction<br />

to this, <strong>the</strong>re are s spaces with 2. > 1 which have <strong>the</strong> RNP (and thus <strong>in</strong> particular do<br />

not conta<strong>in</strong> a subspace isomorphic to c0). For a discussion <strong>of</strong> s spaces see [40].<br />

There are a cont<strong>in</strong>uum <strong>of</strong> isomorphism classes <strong>of</strong> separable s spaces. The simplest<br />

example (besides el and L 1(0, 1)) is <strong>the</strong> kernel <strong>of</strong> a quotient map T :el --+ L1. The isomorphism<br />

type <strong>of</strong> this kernel turns out not to depend on <strong>the</strong> choice <strong>of</strong> T. This kernel is an<br />

s space which is not isomorphic to a complemented subspace <strong>of</strong> an L l(#) space. For a<br />

discussion <strong>of</strong> s spaces, see [22].<br />

There is also a "local view" <strong>of</strong> <strong>Banach</strong> lattices. A <strong>Banach</strong> space X is said to have<br />

Gordon-Lewis local unconditional structure (GL-l.u.st.) if <strong>the</strong>re is a constant 2. so that<br />

for every f<strong>in</strong>ite dimensional subspace E <strong>of</strong> X <strong>the</strong>re is a space Y with an unconditional basis<br />

and operators: T : E --+ Y, S : Y ~ X so that ST = IE and []TI[ ]]Slluc(Y) 1. Indeed, if X is an order complete lattice, <strong>the</strong>n we saw <strong>in</strong> Section 5 that for any f<strong>in</strong>ite<br />

dimensional subspace E and every ~ > 0 <strong>the</strong>re are vectors {xl ..... Xm } <strong>in</strong> X whose span<br />

conta<strong>in</strong>s E and which is an e-perturbation <strong>of</strong> a sequence <strong>of</strong> disjo<strong>in</strong>tly supported vectors.<br />

S<strong>in</strong>ce a disjo<strong>in</strong>tly supported sequence <strong>in</strong> a lattice is 1-unconditional and <strong>the</strong> bidual <strong>of</strong> a<br />

<strong>Banach</strong> lattice is an order complete lattice, <strong>the</strong> general case follows from <strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong><br />

small perturbations and <strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong> local reflexivity.<br />

It is clear from <strong>the</strong> def<strong>in</strong>ition that s spaces, 1 ~< p ~< ec, have 1.u.st.<br />

The close relationship between 1.u.st. and lattice structure is expressed <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g<br />

result: A <strong>Banach</strong> space X has GL-l.u.st. if and only if X** is isomorphic to a complemented

60 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

subspace <strong>of</strong>a <strong>Banach</strong> lattice. Indeed, s<strong>in</strong>ce by <strong>the</strong> def<strong>in</strong>ition a complemented subspace <strong>of</strong><br />

a space with GL-l.u.st. has GL-l.u.st., it follows that if X** is a complemented subspace<br />

<strong>of</strong> a lattice it has GL-l.u.st. By <strong>the</strong> pr<strong>in</strong>ciple <strong>of</strong> local reflexivity we deduce that also X has<br />

GL-l.u.st.<br />

To prove <strong>the</strong> o<strong>the</strong>r direction we use ultraproducts. Assume X has GL-l.u.st. with some<br />

constant )~. For every f<strong>in</strong>ite dimensional subspace E <strong>of</strong> X <strong>the</strong>re are a space YE with 1-<br />

unconditional basis and operators Tt; :E --+ Yt; and St; :Y t; --+ X with l[ Tt; 11 ~< 1, liSt; 11

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 61<br />

whe<strong>the</strong>r <strong>the</strong> disk algebra has <strong>the</strong> UAE As a matter <strong>of</strong> fact, <strong>the</strong> only general class <strong>of</strong> spaces<br />

o<strong>the</strong>r than L p spaces which are known to have <strong>the</strong> UAP are <strong>the</strong> reflexive Orlicz spaces9<br />

10. Some special classes <strong>of</strong> operators<br />

In this section we shall discuss some special classes <strong>of</strong> l<strong>in</strong>ear operators between <strong>Banach</strong><br />

spaces and <strong>the</strong>ir relation to <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces. We shall also discuss some<br />

o<strong>the</strong>r topics <strong>in</strong> operator <strong>the</strong>ory which are relevant to <strong>the</strong> study <strong>of</strong> <strong>the</strong> structure <strong>of</strong> <strong>Banach</strong><br />

spaces.<br />

Most <strong>of</strong> <strong>the</strong> classes <strong>of</strong> operators we consider have <strong>the</strong> ideal property, mean<strong>in</strong>g that <strong>the</strong><br />

operators from a fixed X <strong>in</strong>to a fixed Y hav<strong>in</strong>g this property form a l<strong>in</strong>ear space and that<br />

whenever T : X --+ Y belongs to this class <strong>the</strong>n for every bounded U : Z --+ X and V : Y --+<br />

W <strong>the</strong> operator V T U belongs to this class. The three most elementary operator ideals are<br />

<strong>the</strong> bounded operators, <strong>the</strong> compact operators, and <strong>the</strong> weakly compact operators. S<strong>in</strong>ce<br />

B(X, Y), K(X, Y), and WK(X, Y) are all <strong>Banach</strong> spaces under <strong>the</strong> operator norm, <strong>the</strong><br />

operator norm is <strong>the</strong> natural norm to use for operators <strong>in</strong> <strong>the</strong>se classes. A <strong>Banach</strong> space<br />

X has <strong>the</strong> Dunford-Pettis (DP) property provided that every weakly compact operator<br />

with doma<strong>in</strong> X maps weakly compact sets <strong>in</strong>to norm compact sets. It is clear that if T E<br />

WK(X, Y), S E WK(Y, Z), and Y has <strong>the</strong> DP property, <strong>the</strong>n ST is a compact operator.<br />

In particular, if X has <strong>the</strong> DP property and P is a weakly compact projection on X, <strong>the</strong>n<br />

p = p2 is compact. This means that <strong>the</strong> only complemented reflexive subspaces <strong>of</strong> X are<br />

<strong>the</strong> f<strong>in</strong>ite dimensional ones.<br />

A subset <strong>of</strong> a <strong>Banach</strong> space is relatively weakly compact if and only if it is relatively<br />

weakly sequentially compact. Therefore, a space X has <strong>the</strong> DP property if and only if every<br />

weakly compact operator with doma<strong>in</strong> X maps sequences which converge weakly to zero<br />

<strong>in</strong>to sequences which converge <strong>in</strong> norm to zero.<br />

The follow<strong>in</strong>g is an elegant characterization <strong>of</strong> spaces hav<strong>in</strong>g <strong>the</strong> DP property. X has<br />

<strong>the</strong> DPproperty if and only if whenever {Xn}n~__l <strong>in</strong> X tends weakly to 0 and {x*}n~=l <strong>in</strong> X*<br />

tends weakly to 0 <strong>the</strong> sequence <strong>of</strong> scalars {x* (Xn) }n~__l tends to O. Indeed, assume that X has<br />

~< 11)<br />

DP and x n > 0 9 The operator T" X ~ co def<strong>in</strong>ed by Tx -- (x* 1 (x), x 2 *(x) .... ) is a weakly<br />

compact operator (it is easier to check that T* is weakly compact). Hence, if Xn --+ 0<br />

weakly, Ix*(xn)[ ~< IITxnl[ ~ 0. Conversely, assume <strong>the</strong> condition on <strong>the</strong> sequences is<br />

satisfied, T:X --+ Y is weakly compact, and x~ w> 0. If II Txn IIr 0 we may assume, by<br />

pass<strong>in</strong>g to a subsequence, that II Txn II >~ 6 for some 6 > 0 and all n. Let y* E X* be such<br />

that y*(Txn) -- IlTxn[I and IlYn* l[ - 1 for all n. S<strong>in</strong>ce T* is weakly compact as well we may<br />

assume (pass<strong>in</strong>g aga<strong>in</strong> to a subsequence if needed) that T* y*<br />

0- lim(T*y* - x*)(Xn) --limy*(Txn) --lim IITxn II<br />

n /7 n<br />

to<br />

> x* for some x*. Then<br />

a contradiction.<br />

It follows from <strong>the</strong> criterion we just proved that X has <strong>the</strong> DP property if X* has this<br />

property. (The converse is false: ( Y~n=l ~)1 has <strong>the</strong> DP property but ( Y~m= er 1 e2) n oo conta<strong>in</strong>s<br />

a complemented copy <strong>of</strong> ~2 and hence fails <strong>the</strong> DP property.) It is also clear that a

62 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

complemented subspace <strong>of</strong> a space with <strong>the</strong> DP property has <strong>the</strong> same property. We shall<br />

verify shortly that C(K) spaces have <strong>the</strong> DP property. Consequently we get that: All E~<br />

and s spaces have <strong>the</strong> DP property.<br />

To show that C(K) has <strong>the</strong> DP property let {xn},~__l be a sequence <strong>in</strong> <strong>the</strong> unit ball <strong>of</strong><br />

C(K) so that x,(t) --+ 0 for every t 6 K. Let {lZn},e~=l be a sequence <strong>in</strong> <strong>the</strong> unit ball <strong>of</strong><br />

C(K)* which tends weakly to 0. All <strong>the</strong> #, can be considered as elements <strong>in</strong> L1 (#) where<br />

-- ZnCX~=l I#, 1/2 n. By <strong>the</strong> analysis done <strong>in</strong> Section 4 <strong>of</strong> sets <strong>in</strong> Ll (#) which have weakly<br />

compact closure it follows that for every E > 0 <strong>the</strong>re is a & > 0 so that whenever E is a<br />

Borel set <strong>in</strong> K with #(E) < 6 <strong>the</strong>n I#nl(E) < ~ for all n. By Egor<strong>of</strong>f's <strong>the</strong>orem <strong>the</strong>re is<br />

a set E with #(E) < & so that IXn(t)[

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 63<br />

It follows from this observation that <strong>the</strong> sum <strong>of</strong> two strictly s<strong>in</strong>gular operators is strictly<br />

s<strong>in</strong>gular and thus <strong>the</strong> class <strong>of</strong> strictly s<strong>in</strong>gular operators has <strong>the</strong> ideal property. It is also<br />

easy to check that SS(X, Y) is complete <strong>in</strong> <strong>the</strong> operator norm.<br />

Two <strong>Banach</strong> spaces X and Y are called totally <strong>in</strong>comparable if <strong>the</strong>re is no <strong>in</strong>f<strong>in</strong>ite dimensional<br />

space Z which is isomorphic to subspaces <strong>of</strong> both X and Y. If every operator<br />

from X to Y is strictly s<strong>in</strong>gular (and <strong>in</strong> particular if X and Y are totally <strong>in</strong>comparable)and<br />

Z is a complemented subspace <strong>of</strong> X • Y <strong>the</strong>n <strong>the</strong>re is an automorphism T <strong>of</strong> X 9 Y so that<br />

T Z is <strong>of</strong> <strong>the</strong> form T Z = Xo 9 Yo where Xo (respectively Yo) is a complemented subspace<br />

<strong>of</strong> X (respectively Y). This result is not as simple as <strong>the</strong> preced<strong>in</strong>g results <strong>in</strong> this section.<br />

Its pro<strong>of</strong> can be found <strong>in</strong> [ 11, 2.c. 13].<br />

Ano<strong>the</strong>r class <strong>of</strong> operators is that <strong>of</strong> Fredholm operators. An operator T:X --+ Y is<br />

called a Fredholm operator if or(T) := dimker T < c~ and TX is closed with/~(T) :=<br />

dimY/TX < cx~. The number i(T) = or(T) -/3(T) is called <strong>the</strong> <strong>in</strong>dex <strong>of</strong> T and has a<br />

significant role <strong>in</strong> many applications. As a matter <strong>of</strong> fact i (T) can be def<strong>in</strong>ed also for non-<br />

Fredholm operators provided TX is closed and at least one <strong>of</strong> <strong>the</strong> numbers or(T) or/3(T)<br />

is f<strong>in</strong>ite (<strong>in</strong> this case <strong>the</strong> <strong>in</strong>dex can be <strong>of</strong> course ei<strong>the</strong>r +cxz or -cx~). The class <strong>of</strong> Fredholm<br />

operators is not closed under addition or composition. However if T1 and T2 are Fredholm<br />

operators so is TiT2 (provided it is properly def<strong>in</strong>ed) and i(T1T2) = i(T1) + i(T2). Also<br />

i(T*) = -i (T). These facts are simple exercises <strong>in</strong> l<strong>in</strong>ear algebra. Somewhat harder to<br />

prove is that Fr(X, Y) is preserved by strictly s<strong>in</strong>gular perturbations. If T, S : X --+ Y with<br />

T Fredholm and S strictly s<strong>in</strong>gular <strong>the</strong>n T + S is also Fredholm with i(T + S) -- i(T) (see<br />

[11, 2.c.9]).<br />

An operator T'X --+ Y is called absolutely summ<strong>in</strong>g if whenever EiC~=l xi converges<br />

unconditionally <strong>in</strong> X <strong>the</strong> series Ei~l Txi converges absolutely. From <strong>the</strong> closed graph<br />

<strong>the</strong>orem one deduces easily that T is absolutely summ<strong>in</strong>g if and only if <strong>the</strong>re is a constant<br />

C so that for any choice <strong>of</strong> {xi }<strong>in</strong>=l <strong>in</strong> X<br />

EII Txi ]l ~ C sup I x*(xi)]" I[X* II ~ 1 .<br />

i---1<br />

i--1<br />

More generally, for every 0 < p < cx~ we can def<strong>in</strong>e <strong>the</strong> class <strong>of</strong> p-summ<strong>in</strong>g operators as<br />

those operators for which <strong>the</strong>re is a constant C so that for all choices <strong>of</strong> {xi n }i=1 <strong>in</strong> X<br />

IlTxill p

- sup<br />

64 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

Without fur<strong>the</strong>r mention we shall henceforth assume that p ~> 1 when discuss<strong>in</strong>g p-<br />

summ<strong>in</strong>g operators. However, <strong>the</strong>re are important applications to <strong>Banach</strong> space <strong>the</strong>ory <strong>of</strong><br />

p-summ<strong>in</strong>g operators for 0 < p < 1 (see, e.g., <strong>the</strong> article [32]), and much <strong>of</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong><br />

p-summ<strong>in</strong>g operators, 1 ~< p < oo, goes over to <strong>the</strong> range 0 < p < 1. In particular, <strong>the</strong>re<br />

is a version <strong>of</strong> <strong>the</strong> Pietsch factorization <strong>the</strong>orem (discussed below when 1 ~< p < oo) for<br />

p-summ<strong>in</strong>g operators, 0 < p < 1 (see [20, Theorem 9.1]).<br />

Notice that <strong>the</strong> supremum on <strong>the</strong> right side <strong>of</strong> (28) is <strong>the</strong> norm <strong>of</strong> <strong>the</strong> operator from s<br />

to X def<strong>in</strong>ed by ek ~ xk (notice that p ~> 1 is needed for this). Thus top(T) can also be<br />

def<strong>in</strong>ed by<br />

Zcp(T) -<br />

[[TVenl] p " V " g.p, --~ X, []VI] ~< 1 . (29)<br />

Equation (29) implies that 7rp(T) is <strong>the</strong> supremum <strong>of</strong> rcp(TV) as V varies over <strong>the</strong> norm<br />

one operators from s <strong>in</strong>to X.<br />

In contrast to <strong>the</strong> classes <strong>of</strong> weakly compact or strictly s<strong>in</strong>gular operators, <strong>the</strong> class <strong>of</strong> p-<br />

summ<strong>in</strong>g operators is determ<strong>in</strong>ed by <strong>the</strong> behavior <strong>of</strong> T on <strong>the</strong> f<strong>in</strong>ite dimensional subspaces<br />

<strong>of</strong> X. Because <strong>of</strong> <strong>the</strong> quantitative nature <strong>of</strong> its def<strong>in</strong>ition <strong>the</strong> notion <strong>of</strong> p-summ<strong>in</strong>g norm<br />

plays also an important role <strong>in</strong> <strong>the</strong> study <strong>of</strong> f<strong>in</strong>ite dimensional spaces.<br />

If K is a compact Hausdorff space and/z a regular probability measure on K <strong>the</strong>n <strong>the</strong><br />

identity operator Ip from C(K) <strong>in</strong>to Lp(#) is easily seen to be a p-summ<strong>in</strong>g operator<br />

with p-summ<strong>in</strong>g norm one. It turns out that this simple example is <strong>the</strong> prototype <strong>of</strong> general<br />

p-summ<strong>in</strong>g operators. This is <strong>the</strong> content <strong>of</strong> <strong>the</strong> Pietsch factorization <strong>the</strong>orem. Let X be a<br />

subspace <strong>of</strong> C (K). An operator T : X --+ Y is p-summ<strong>in</strong>g, 1 Xp<br />

Y

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 65<br />

(For p -- 2, <strong>the</strong> operator S can be extended to an operator from L p (#) <strong>in</strong>to Y, but for p :/: 2<br />

this is not <strong>in</strong> general <strong>the</strong> case.)<br />

To prove <strong>the</strong> Pietsch factorization <strong>the</strong>orem, assume that 7rp(T) = 1. Consider <strong>the</strong> follow<strong>in</strong>g<br />

subsets <strong>of</strong> C(K)<br />

G = {f e C(K): sup{f(t)" t 6 K} < 1},<br />

F = conv{f E C(K): f(-)- Ix(.)l z', IITxll- 1}.<br />

The sets F and G are convex with G open and, s<strong>in</strong>ce 7rp(T) = 1, F M G = 0. By <strong>the</strong><br />

separation <strong>the</strong>orem <strong>the</strong>re is a regular signed measure # on K and a positive )~ so that<br />

ffd#)~ for f EF. S<strong>in</strong>ce wheneverg~< f and fEG<br />

also g E G it follows that <strong>the</strong> measure/z has to be positive so after normalization we may<br />

assume that it is a probability measure. Also s<strong>in</strong>ce G conta<strong>in</strong>s <strong>the</strong> open unit ball <strong>of</strong> C(K)<br />

we must have )~ >~ 1. Hence for every x ~ X, IITx[I p /.... If <strong>the</strong> sequence is <strong>in</strong>f<strong>in</strong>ite, <strong>the</strong>n<br />

necessarily )~n(T) --+ O. For <strong>the</strong> <strong>Banach</strong> space ~2, <strong>the</strong> spectral <strong>the</strong>orem for compact op-

66 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

erators says that if T ~ K(H, H), <strong>the</strong>n <strong>the</strong>re is an orthonormal basis {Xn}nC~ 1 SO that <strong>the</strong><br />

self-adjo<strong>in</strong>t operator T*T is represented as<br />

T*Tx -- Z )~n(T*T)(x,xn)Xn (31)<br />

nEP<br />

with )~n(T*T) > 0 for n 6 P and T*Txn- 0 for n r P.<br />

Now if X is a <strong>Banach</strong> space with a 1-symmetric basis S- {en}n~__l , let S(62) be those<br />

compact operators on 62 for which ~ ~/)~n(T*T) en converges <strong>in</strong> X, and, for T 6 S(62), set<br />

Then (S(62), o's) is a <strong>Banach</strong> space (verify<strong>in</strong>g <strong>the</strong> triangle <strong>in</strong>equality is a bit tricky). It is<br />

less difficult to check that S(62) satisfies <strong>the</strong> unitary ideal property<br />

crs(UTV)

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 67<br />

orthonormal basis {en}n~__| and )~, ~> 0 so that Ten -- )~nen for every n and 1 -- (re(T) --<br />

~nC~=l)~n .2 Let {~n}~_|_ be a Rademacher sequence. Us<strong>in</strong>g Kh<strong>in</strong>tch<strong>in</strong>e's <strong>in</strong>equality (1) we<br />

get<br />

sup I(xi,x)[" Ilxll ~< 1 , ~> sup xi, Z-l-)~jej<br />

i--1 -+- i--1 j--1<br />

>>. ~ E ~_~)~j~j (xi, e~) >1 el ~_~ I,~j (xi, ej)l 2 -- A1 II Zxi II.<br />

i=1 j--1 i=1 j=l i=1<br />

This shows that 7rl (T) ~< A11 crz(T).<br />

To show <strong>the</strong> converse, it is enough to check that if T 6//p(~2, ~2) with 2

68 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

norm at most one when considered as an operator from s to s Let H be a Hilbert space.<br />

Then IIA @ In II ~< KG as an operator from s (H) to s (H), where<br />

(A | IH)(yl ..... Yn) "= aljyj ..... anjyj<br />

j=l<br />

for (Yl ..... Yn) <strong>in</strong> s<br />

To prove Gro<strong>the</strong>ndieck's <strong>in</strong>equality fix n and a matrix A- (aij)<strong>in</strong>, j=l with [IAll "-<br />

IIAs ---> s 1 as above, and put !A! := [IA | IH's ---> s We have to<br />

f<strong>in</strong>d an estimate on !A! <strong>in</strong>dependent <strong>of</strong> n and A. Note first that for all ui and vj <strong>in</strong> a Hilbert<br />

space<br />

j=l<br />

aij (ui, v j)<br />

i,j=l<br />

~< !A! max Ilui II max Ilvj II.<br />

i j<br />

S<strong>in</strong>ce all Hilbert spaces are created equal, we can use for H any (<strong>in</strong>f<strong>in</strong>ite dimensional or<br />

even 2n-dimensional) Hilbert space. For <strong>the</strong> purpose <strong>of</strong> prov<strong>in</strong>g Gro<strong>the</strong>ndieck's <strong>in</strong>equality,<br />

some Hilbert spaces are more equal than o<strong>the</strong>rs! We use for H <strong>the</strong> subspace <strong>of</strong> L2(0, 1)<br />

mentioned <strong>in</strong> Section 4 consist<strong>in</strong>g just <strong>of</strong> functions hav<strong>in</strong>g a Gaussian distribution with<br />

mean 0. To prove Gro<strong>the</strong>ndieck's <strong>in</strong>equality it is enough to consider norm one vectors<br />

{Xi}<strong>in</strong>=l and {YJ}j=I <strong>in</strong> H. Given 0 < 6 < 1/2 <strong>the</strong>re is an M -- M(6) so that for any norm<br />

one function f <strong>in</strong> H, Ilf - f M 112 --" • where<br />

fM (t) - ] f (t) if If(t)l ~< M,<br />

M sign f(t) if If(t)[ > M.<br />

[<br />

Note that by our assumption on <strong>the</strong> matrix A, for any choice <strong>of</strong> functions f/ and g j <strong>in</strong><br />

L2 (0, 1) which are uniformly bounded by M,<br />

Zaij(fi,gj) f01 Z aij fi(t)gj(t)<br />

9 .<br />

i,j t,j<br />

dt ~< M 2.<br />

Hence<br />

Z aij (xi, yj )<br />

i,j<br />

Zai,j(XiM, y M}<br />

i,j<br />

+ Zaij(xi, Yj -- Y M)<br />

i,j<br />

+ Z aij(xi- x M, yM)<br />

i,j<br />

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 69<br />

<strong>in</strong>equality. With <strong>the</strong> optimal choice <strong>of</strong> 6 this pro<strong>of</strong> yields that Kc < 8.69 (or KG < 8.55 if<br />

one exercises more care <strong>in</strong> <strong>the</strong> computation), but it is known that KG < 1.79. Incidentally,<br />

although <strong>the</strong> best value for Kc is unknown, it is known that for complex scalars it is smaller<br />

than for real ones.<br />

The first application <strong>of</strong> Gro<strong>the</strong>ndieck's <strong>in</strong>equality is: Every bounded l<strong>in</strong>ear operator<br />

T from s to ~2 is absolutely summ<strong>in</strong>g and rrl(T) ~ KGIITII. Indeed, let {ej}7_ 1 be <strong>the</strong><br />

unit vector basis for s and let ui -- Y'~jm= 1 aijej be n vectors <strong>in</strong> s for some m so that<br />

E<strong>in</strong>=_ , m <strong>of</strong> scalars <strong>of</strong><br />

1]x (ui)l ~< 1 for every unit vector x* <strong>in</strong> s For any choice {Sj}j= 1<br />

absolute value

70 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

.....<br />

def<strong>in</strong>e <strong>in</strong> g~c ( H) <strong>the</strong> vector 2 "- (yl ym).Thenllxlle~(.)<br />

__ ( n 1/2<br />

II Y~k=l Ixkl 2) I1~ ~< 1.<br />

Similarly,<br />

k=l<br />

ITxkl 2<br />

-II T o<br />

and this last quantity is at most KGIITII by (<strong>the</strong> conceptual form <strong>of</strong>) Gro<strong>the</strong>ndieck's<br />

<strong>in</strong>equality. This gives (33) when X- ~, Y- g~n and hence also when X = C(K),<br />

Y = L1 (#).<br />

Suppose now that X and Y are general <strong>Banach</strong> lattices and T :X --+ Y has norm one.<br />

Let x l ..... Xn be <strong>in</strong> X with II (~i=1 n Ixil 2) 1/2 II- 1 and set u -- (y~<strong>in</strong>__l Ixil2) 1/2 S<strong>in</strong>ce we<br />

are <strong>in</strong>terested only <strong>in</strong> estimat<strong>in</strong>g II(Y~i~I I Txi 12) 1/2ll, we can assume by replac<strong>in</strong>g X by<br />

<strong>the</strong> (separable) sublattice generated by x l ..... Xn that X is separable. The space T X is<br />

<strong>the</strong>n separable, so we can similarly assume that Y is separable. Given any ~ > 0, as seen <strong>in</strong><br />

Section 5 <strong>the</strong>re is a strictly positive functional y* E Y* with<br />

y* ITxil 2<br />

i=1 i:,<br />

1/2<br />

and IlY*II ~ 1 -q- ~. As mentioned <strong>in</strong> Section 5, <strong>the</strong> space Xu is lattice isometric to a C(K)<br />

space and Yy, is lattice isometric to an L l(#) space. The natural lattice homomorphisms<br />

J1 : Xu --+ X, J2 : Y --+ Yy* satisfy II J1 II = Ilu II = 1 and II J2 II = IlY* II ~< 1 + ~. We can <strong>the</strong>n<br />

apply (33) to <strong>the</strong> operator J2 T J1 to obta<strong>in</strong> that<br />

1/2<br />

i=1<br />

Irxil 2<br />

Yv*<br />

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 71<br />

<strong>in</strong>f<strong>in</strong>ite dimensional). Note that ~2 is isometric to a subspace <strong>of</strong> L1 (0, 1) (see Section 4)<br />

and thus ~ -- ~2 is also a quotient space <strong>of</strong> C(0, 1) because ~2 is reflexive and C(0, 1),<br />

considered as a subspace <strong>of</strong> L~ (0, 1) -- L 1(0, 1)*, determ<strong>in</strong>es <strong>the</strong> norm <strong>of</strong> L1 (0, 1). (If Y<br />

is a reflexive subspace <strong>of</strong> a <strong>Banach</strong> space X and Z is a norm closed subspace <strong>of</strong> X* which<br />

determ<strong>in</strong>es <strong>the</strong> norm <strong>of</strong> X, <strong>the</strong>n <strong>the</strong> restriction operator z* ~ z~y is a quotient mapp<strong>in</strong>g<br />

from Z onto Y*.)<br />

We will next give a glimpse <strong>in</strong>to <strong>the</strong> connections <strong>of</strong> p-summ<strong>in</strong>g operators to <strong>the</strong> <strong>the</strong>ory<br />

<strong>of</strong> f<strong>in</strong>ite dimensional spaces discussed <strong>in</strong> Section 8. If X is <strong>the</strong> space g~ <strong>the</strong>n 7f2 (Ix) =<br />

s<strong>in</strong>ce this is <strong>the</strong> Hilbert-Schmidt norm <strong>of</strong> <strong>the</strong> identity <strong>in</strong> g~. It is quite surpris<strong>in</strong>g that: For<br />

any X with dim X = n, re2 (Ix) = ~/-ff. To see this take any norm one operator V from ~2<br />

<strong>in</strong>to X. By project<strong>in</strong>g onto <strong>the</strong> orthogonal complement <strong>of</strong> <strong>the</strong> kernel <strong>of</strong> V, we can factor V<br />

as V1P where VI"~ n -+ X, P's --+ ~n, II P II - 1, II V1 II - II V II, and m ~< n. Then<br />

yr2(Ix g) ~ IIIxll IIV~ll~2(IeT)llPII ~ v/-n,<br />

Tak<strong>in</strong>g <strong>the</strong> supremum over all such V gives Yr2 (Ix) ~ ~/-n.<br />

To prove <strong>the</strong> o<strong>the</strong>r <strong>in</strong>equality, we get from <strong>the</strong> Pietsch factorization <strong>the</strong>orem a probability<br />

measure # on Bx. and T :L2(#) --+ X so that TI21x = Ix and IITII = 7rz(Ix), where as<br />

usual I2:C(Bx.) --+ L2(#) is <strong>the</strong> formal identity and X is canonically embedded <strong>in</strong>to<br />

C(Bx.). The space X2 := I2X is an n-dimensional Hilbert space and IzT is <strong>the</strong> identity<br />

on X2. Hence<br />

--- 7F2(Ix2) ~ 7r2(I2)llTII = 7r2(Ix).<br />

As a consequence <strong>of</strong> <strong>the</strong> preced<strong>in</strong>g we deduce: The projection constant <strong>of</strong>an n-dimensional<br />

space X is at most v/if; that is, whenever Y conta<strong>in</strong>s X <strong>the</strong>re is a projection P from Y onto<br />

X with I[ P [[ ~< v/-~. Indeed, s<strong>in</strong>ce fez(Ix) = V/if, <strong>the</strong> Pietsch factorization <strong>the</strong>orem yields<br />

that <strong>the</strong> identity operator on X can be represented as X J > Loo(/z) 12> L2(#) v > X,<br />

with J an isometry, 12 <strong>the</strong> formal identity, and II vii- v/if, S<strong>in</strong>ce L~(#) is 1-<strong>in</strong>jective<br />

<strong>the</strong> operator J can be extended to a norm one operator T from Y <strong>in</strong>to L~(#). The operator<br />

P :-- V12 T is a projection from Y onto X with norm at most v/ft. Notice also that<br />

<strong>the</strong> factorization used above gives ano<strong>the</strong>r pro<strong>of</strong> <strong>of</strong> <strong>the</strong> result proved <strong>in</strong> Section 8 that<br />

d(X, ~) ~ x/~.<br />

The estimate <strong>of</strong> x/~ for <strong>the</strong> projection constant <strong>of</strong> an n-dimensional space is essentially<br />

sharp. This is discussed <strong>in</strong> [33].<br />

The p-<strong>in</strong>tegral operators form a class <strong>of</strong> operators which are closely related to <strong>the</strong> p-<br />

summ<strong>in</strong>g operators. An operator T :X --~ Y is said to be p-<strong>in</strong>tegral, 1 ~ p ~ cx) (<strong>in</strong> symbols<br />

T E Zr(X, Y)), provided that <strong>the</strong> composition Jr T <strong>of</strong> T with <strong>the</strong> canonical embedd<strong>in</strong>g<br />

Jy :Y --~ Y** factors through <strong>the</strong> formal identity I~,p : L~(#) ~ Ln(#) for some<br />

probability measure #:<br />

L~(#)<br />

I~, p<br />

> Lp(#)<br />

AI T JY ~B<br />

X > Y > Y**<br />

(34)

72 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

The p-<strong>in</strong>tegral norm ip(T) is <strong>the</strong>n def<strong>in</strong>ed to be <strong>the</strong> <strong>in</strong>fimum over all such factorizations<br />

<strong>of</strong> [[ A [[ [[ B [[. By tak<strong>in</strong>g ultraproducts one sees that this <strong>in</strong>fimum is really a m<strong>in</strong>imum. The<br />

space (Zp(X, Y), ip) is easily seen to be a <strong>Banach</strong> space and ip satisfies <strong>the</strong> ideal property<br />

ip(STU) ~ []S[[ip(T)[[U[[.<br />

If T is <strong>in</strong> Zp(X, Y) and X is a subspace <strong>of</strong> C(K), with K a compact Hausdorff space,<br />

<strong>the</strong>n <strong>the</strong>re is a probability measure v on K and an operator S:Lp(v) ---+ Y** with [[S[[ ----<br />

ip(T) which makes <strong>the</strong> follow<strong>in</strong>g diagram commute:<br />

C(K)<br />

Ip<br />

> Lp(v)<br />

X<br />

T<br />

> Y<br />

Jr<br />

> Y**<br />

(35)<br />

Indeed, if (34) holds, A can be extended to an operator A" C(K) --+ L~(#) because<br />

L~(#) is 1-<strong>in</strong>jective. By <strong>the</strong> Pietsch factorization <strong>the</strong>orem <strong>the</strong>re is a probability measure<br />

v on K so that for each x E C (K),<br />

[[BI~,pAx[] ~ 7rp(Bl~,pA)<br />

[x[ p dv<br />

and <strong>the</strong> desired conclusion follows from<br />

7rp(BIc~,pA) ~ IIBllTrp(l~,p)llS, II =<br />

IIBlllIAII.<br />

It is evident that zrp(T) ~ ip(T) and from <strong>the</strong> Pietsch factorization <strong>the</strong>orem it follows<br />

that Zcp(T) = ip(T) if <strong>the</strong> doma<strong>in</strong> <strong>of</strong> T is a C(K) space. As was mentioned implicitly <strong>in</strong><br />

<strong>the</strong> discussion <strong>of</strong> 2-summ<strong>in</strong>g operators, 7r2(T) -- i2(T) for any operator. One reason for<br />

def<strong>in</strong><strong>in</strong>g p-<strong>in</strong>tegral via a factorization <strong>of</strong> Jy T ra<strong>the</strong>r than T is that this forces i p(T) --<br />

ip(T**) (use <strong>the</strong> fact that a dual space is norm one complemented <strong>in</strong> its bidual).<br />

The 1-<strong>in</strong>jectivity <strong>of</strong> C(K)** gives that a p-summ<strong>in</strong>g operator T <strong>in</strong>to a C(K) space is<br />

p-<strong>in</strong>tegral with ip(T) = 7rp(T). The equality il (T*) = il (T) follows from <strong>the</strong> observation<br />

that <strong>the</strong> adjo<strong>in</strong>t I~ <strong>of</strong> 11 "C(K) --+ Ll(v) is l~,l'L~(v) --+ Ll(v) followed by <strong>the</strong> identification<br />

<strong>of</strong> L1 (v) with <strong>the</strong> norm one complemented subspace <strong>of</strong> C(K)* consist<strong>in</strong>g <strong>of</strong> <strong>the</strong><br />

f<strong>in</strong>ite signed measures which are absolutely cont<strong>in</strong>uous with respect to v.<br />

For o<strong>the</strong>r values <strong>of</strong> p, <strong>the</strong> adjo<strong>in</strong>t <strong>of</strong> a p-<strong>in</strong>tegral operator need not be strictly s<strong>in</strong>gular<br />

(see [9, 5.12]) and hence need not be q-summ<strong>in</strong>g for any q < c~.<br />

For each 1 ~< p ~< cx~, p -r 2, <strong>the</strong>re exist p-summ<strong>in</strong>g operators which are not p-<strong>in</strong>tegral<br />

(see [9, 5.13]). The case p -- 1 is particularly easy to deduce from <strong>the</strong> <strong>the</strong>ory we have presented.<br />

We saw that every operator T :~1 --+ ~2 is 1-summ<strong>in</strong>g. If T :~1 --+/~2 is 1-<strong>in</strong>tegral,<br />

A I~, 1 B<br />

<strong>the</strong>n T has a factorization ~1 > L~ (#) > L1 (/z) > ~2. But <strong>the</strong>n B and also I~, 1 are<br />

1-summ<strong>in</strong>g, hence B I~, l, whence also B I~, 1A = T, are compact (use <strong>the</strong> fact that L 1 (/z)<br />

has <strong>the</strong> DP property).<br />

For p = c~ and Y reflexive <strong>the</strong> <strong>in</strong>f<strong>in</strong>ity <strong>in</strong>tegral operators from X to Y are exactly those<br />

which factor through some L~(#) space (with <strong>the</strong> <strong>in</strong>tegral norm equal to <strong>the</strong> best factorization).<br />

In particular for X reflexive <strong>the</strong> <strong>in</strong>f<strong>in</strong>ity <strong>in</strong>tegral norm <strong>of</strong> <strong>the</strong> identity <strong>of</strong> X is

- 1/x<br />

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 73<br />

f<strong>in</strong>ite if and only if X is f<strong>in</strong>ite dimensional (and is equal to <strong>the</strong> projection constant <strong>of</strong> X <strong>in</strong><br />

that case). Thus also for p -- cx~ it is evident that summ<strong>in</strong>g and <strong>in</strong>tegral norms can be very<br />

different.<br />

The ma<strong>in</strong> reason for <strong>in</strong>troduc<strong>in</strong>g p-<strong>in</strong>tegral operators is that <strong>the</strong>y are needed for <strong>the</strong><br />

duality <strong>the</strong>ory <strong>of</strong> lip(X, Y). For simplicity, we restrict to <strong>the</strong> case where X and Y are<br />

f<strong>in</strong>ite dimensional. Follow<strong>in</strong>g <strong>the</strong> notation used <strong>in</strong> Section 8, for f<strong>in</strong>ite dimensional X,<br />

Y and ot a norm on B(X, Y) we represent <strong>the</strong> dual <strong>of</strong> (B(X, Y), or) as (B(Y, X),ot*),<br />

where <strong>the</strong> pair<strong>in</strong>g is given by (S, T) = trace TS (= traceST). Then for all 1 ~< p ~< cx~,<br />

Hp(X, Y)* = Zp,(Y, X) when X and Y are f<strong>in</strong>ite dimensional. This just means that for<br />

each S E B(Y,X), ip,(S) = sup{traceTS: T E B(X,Y),rcp(T) L1 (#)<br />

A'I S ~B,<br />

Y >X<br />

(37)<br />

one has for E > 0 an operator A~ :Y --~ L~(#) with [[A1 - Aell < ~ so that A~Y is conta<strong>in</strong>ed<br />

<strong>in</strong> <strong>the</strong> simple functions. This gives a factorization <strong>of</strong> B11~,l A~ <strong>of</strong> <strong>the</strong> form (36)<br />

with IIBIIIIAIIIIAII ~< IIB~ II III~,~ IIIIA~ II. Sett<strong>in</strong>g N := dim Y, we get A[(S - BII~,IAE)

74 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

and<br />

The operator S** B Ip satisfies<br />

ir(S**BIp) -- 7cr(S**BIp)

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 75<br />

space Z can be replaced with <strong>the</strong> algebraic sum Z(X) "-- Xo + X1 C Z topologized with<br />

<strong>the</strong> (complete) norm<br />

Ilxll - <strong>in</strong>f[ Ilxollto] -+- Ilxl Ill1]" x - xo -4- Xl, xo E Xo, xl ~ xl },<br />

where I1" I1[i1 is <strong>the</strong> norm <strong>of</strong> Xi. We always take for Z <strong>the</strong> space Z(X). The space A(X) "--<br />

Xo N X1 also plays a role <strong>in</strong> <strong>the</strong> <strong>the</strong>ory; it is naturally normed by <strong>the</strong> (complete) norm<br />

IIx II - IIx Ilto] v IIx lit 1].<br />

Thus we have for i -- 0, 1 <strong>in</strong>clusions<br />

Z(X)<br />

with Ilni II ~ 1, IIJi II ~ 1. Any <strong>Banach</strong> space X satisfy<strong>in</strong>g A(X) C X C Z(X) with both<br />

<strong>in</strong>clusions cont<strong>in</strong>uous is called an <strong>in</strong>termediate space between Xo and X1 (or an <strong>in</strong>termediate<br />

space with respect to X). The spaces Xo, X1, A(X), and Z(X) are all <strong>in</strong>termediate<br />

spaces with respect to X.<br />

Given <strong>Banach</strong> couples X -- (Xo, X1), Y -- (Yo, Y1 ), and a l<strong>in</strong>ear mapp<strong>in</strong>g T" Z (X) ----><br />

Z(Y), we write T E B(X, Y) provided Tix i E B(Xi, Yi) for i --0, 1.<br />

If X and Y are <strong>in</strong>termediate spaces with respect to X and Y, respectively, we say that<br />

X and Y are an <strong>in</strong>terpolation pair for X and Y provided that if T E B(X, Y), <strong>the</strong>n TIx<br />

B(X, Y). If always<br />

II T II x, Y ~ II T II go, Yo v II T II x,, Y,<br />

X and Y are said to be an exact <strong>in</strong>terpolation pair for X and Y, where IlSlJx, Y :--<br />

][Slxl]B(X,y ) (if SX r Y <strong>the</strong>n I]Sllx, Y "- oc). F<strong>in</strong>ally, if 0 ~< 0 ~< 1 is such that <strong>the</strong> <strong>in</strong>equality<br />

1-0 0<br />

II T IIx, Y ~ cII TIIxo,Yo. II T IIx,,y,<br />

always holds, <strong>the</strong>n <strong>the</strong> <strong>in</strong>terpolation pair X and Y are said to be <strong>of</strong> exponent 0 (and exact<br />

<strong>of</strong> exponent 0 if C -- 1). When X -- Y and X -- Y, we abbreviate by say<strong>in</strong>g, e.g., that X is<br />

an <strong>in</strong>terpolation space with respect to X provided X and X are an <strong>in</strong>terpolation pair with<br />

respect to X and X.<br />

The most classical realization <strong>of</strong> this abstract set-up occurs <strong>in</strong> <strong>the</strong> scale <strong>of</strong> Lp(#) spaces.<br />

For 1 ~< p0, pl ~< oc, <strong>the</strong> pair (Lpo(#), Lpl (#)) is a <strong>Banach</strong> couple, and for p0/x pl ~<<br />

p ~< P0 v pl <strong>the</strong> space Lp(l z) is an <strong>in</strong>termediate space between Lpo (/z) and Lpj (#). In <strong>the</strong><br />

language <strong>of</strong> <strong>in</strong>terpolation <strong>the</strong>ory <strong>the</strong> Riesz-Thor<strong>in</strong> <strong>in</strong>terpolation <strong>the</strong>orem can be stated as<br />

follows. Let 1

I<br />

76 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

Then for all measures tx and v, (complex) Lp(Ix) and Lq(v) are an exact <strong>in</strong>terpolation<br />

pair <strong>of</strong> exponent 0 with respect to (Lpo(ix), Lpl (Ix)) and (Lqo(V), Lq, (v)). It is formal to<br />

derive from <strong>the</strong> Riesz-Thor<strong>in</strong> <strong>the</strong>orem that its statement rema<strong>in</strong>s true <strong>in</strong> <strong>the</strong> sett<strong>in</strong>g <strong>of</strong> real<br />

scalars as long as P0 ~< pl and q0 ~< ql (or, what is <strong>the</strong> same, if pl ~ p0 and ql

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 77<br />

m<br />

0 with respect to X and Y. To see this, let T 9 B(X, Y), x 9 X[ol, e > 0, and choose<br />

f 9 7-/(X) so that f(O) -- x with IlfllT-t(~) ~< Ilxll[0] + e. Def<strong>in</strong>e g'S --+ S,(Y) by<br />

g(z)-- IlZll%oly ollrll-=<br />

X1, Y1 Tf(z)<br />

-- 0-1 -0<br />

Then g is <strong>in</strong> 7-/(Y), IlgllTt(F) ~< II f llT-t(N), and g(O) -IlTllxo,vollTll x,, Y, Tx so that<br />

1-0<br />

IITxll[ol ~ IITIIxo,YollTIl~<br />

Ilgll~(y)<br />

1-0<br />

78 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

(Lpo (/Z), Lpl (/z))[0] --- Lp(/Z). The argument does use <strong>the</strong> follow<strong>in</strong>g general fact about <strong>the</strong><br />

complex method. For any <strong>Banach</strong> couple X and 0 < 0 < 1, A(X) is dense <strong>in</strong> X[o] (see [4,<br />

4.2.2]).<br />

The identification Lp - (Lpo, Lpl)[O] , ~ 1 = -~-~-- 1-0 _ z- ~-~-, 0 has a generalization to <strong>Banach</strong><br />

lattices. Assume that X0 and X1 are complex <strong>Banach</strong> lattices <strong>of</strong>/z-measurable functions<br />

which are ideals <strong>in</strong> <strong>the</strong> space <strong>of</strong> all/z-measurable functions, 0 < 0 < 1, and set<br />

1 -<br />

S 0 {(sign xoxl)lxoll-~<br />

. .<br />

~ xo ~ S0, Xl E Xl}<br />

0 XOl ._<br />

If Xo is order cont<strong>in</strong>uous <strong>the</strong>n (Xo, Xl)[O] =<br />

X~ -0 XO1 and<br />

Ilxll[0] -<strong>in</strong>f{llx011[0] 1-0 Ilxl II~l] 9 Ixl-Ix011-~ Ix1 I ~ }. (38)<br />

For a pro<strong>of</strong> see Section iv.l.11 <strong>in</strong> [13]. The <strong>in</strong>equality

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 79<br />

Earlier we saw how <strong>the</strong> complex method can be used to construct <strong>the</strong> p-convexification<br />

<strong>of</strong> a <strong>Banach</strong> lattice. The /C-method can also be used to construct <strong>Banach</strong> spaces with<br />

<strong>in</strong>terest<strong>in</strong>g properties. Suppose, for example, that both X0 and X t have a normalized<br />

1-unconditional basis, each <strong>of</strong> which we identify with <strong>the</strong> unit vectors {en}n~=l so that<br />

X0 and X1 are both conta<strong>in</strong>ed <strong>in</strong> co and thus X - (X0, X1) is a <strong>Banach</strong> couple. Let E<br />

be ano<strong>the</strong>r space with a normalized 1-unconditional basis, also denoted by {en}n~ S<strong>in</strong>ce<br />

/C(X, E, a, b) is an exact <strong>in</strong>terpolation space with respect to X, it follows that {en}n~__l is a<br />

1-unconditional basis for/C(X, E, a, b) which moreover is even 1-symmetric if {en}~__l is<br />

1-symmetric <strong>in</strong> both X0 and <strong>in</strong> X1 (one need only check that <strong>the</strong> span <strong>of</strong> {en }n~ is dense <strong>in</strong><br />

/C(X, E, a, b)). Assume now that {en}ncC__l is 1-symmetric <strong>in</strong> both X0 and X1, X0 C X1, and<br />

Y~k-1 ek [l[0] II Ek-1 ekll[l] --+ 0 as n --+ oc. It turns out (see [14, 3.b.4]) that if an<br />

and <strong>the</strong> weight sequence b satisfies bn 1' ec sufficiently quickly, <strong>the</strong>n <strong>the</strong>re are disjo<strong>in</strong>t<br />

f<strong>in</strong>ite subsets Cn <strong>of</strong> N so that <strong>the</strong> mapp<strong>in</strong>g en ~ II lc~ IlK;(N,a,b)lcn extends to an isomor-<br />

phism from E onto a complemented subspace <strong>of</strong>/C(X, E, a, b). This is how one proves<br />

<strong>the</strong> result mentioned <strong>in</strong> Section 3 that a space with an unconditional basis is isomorphic to<br />

a complemented subspace <strong>of</strong> a space which has a symmetric basis.<br />

"Good" properties (such as reflexivity and superreflexivity) possessed by <strong>the</strong> <strong>Banach</strong><br />

spaces X0 and X1 (<strong>of</strong>ten possessed by just one <strong>of</strong> <strong>the</strong> spaces) generally pass to <strong>in</strong>terpolation<br />

spaces between X0 and X l which are obta<strong>in</strong>ed by <strong>the</strong> complex method or by <strong>the</strong> E-method<br />

(at least when <strong>the</strong> weight sequences a, b satisfy some growth conditions and <strong>the</strong> space E<br />

is "nice"). Sometimes <strong>in</strong>terpolation spaces even have a good property which nei<strong>the</strong>r X0<br />

nor X1 possesses. For example, suppose that X0 C X1 (this can be relaxed but is good<br />

enough for applications). If E is reflexive, <strong>the</strong> <strong>in</strong>clusion J" Xo ~ X1 is weakly compact,<br />

O0<br />

}~,n=l an < oc, and bn "~ oc, <strong>the</strong>n/C(X, E, a, b) is reflexive. For a pro<strong>of</strong> when E- ~2,<br />

which is easily modified to cover <strong>the</strong> general case, see [ 15, 2.g. 11 ]. A consequence <strong>of</strong> this<br />

result is <strong>the</strong> follow<strong>in</strong>g factorization <strong>the</strong>orem for weakly compact operators. If T" X ~ X1,<br />

is weakly compact <strong>the</strong>n T factors through a reflexive space; that is, <strong>the</strong>re exists a reflexive<br />

<strong>Banach</strong> space Y and operators A ~ B(X, Y), B ~ B(Y, X1) so that T -- BA. To derive<br />

this factorization <strong>the</strong>orem from <strong>the</strong> <strong>in</strong>terpolation result mentioned above, it suffices to take<br />

Y-/C(X0 X1 ~2 { 2-n }n= ec 1 { 2n } oc 1) where X0 is <strong>the</strong> span <strong>in</strong> X <strong>of</strong> T Bx with T Bx as<br />

<strong>the</strong> unit ball <strong>of</strong> X0. Let A be T, considered as an operator <strong>in</strong>to Y, and let B be <strong>the</strong> formal<br />

<strong>in</strong>clusion from Y <strong>in</strong>to X l. Then A and B are operators and T -- BA.<br />

We turn now to <strong>the</strong> realization <strong>of</strong> <strong>the</strong> /C-method which is used most <strong>of</strong>ten <strong>in</strong> analysis<br />

and is discussed extensively <strong>in</strong> <strong>the</strong> books on <strong>in</strong>terpolation <strong>the</strong>ory we have mentioned.<br />

Given a <strong>Banach</strong> couple X- (Xo, X1), 0 < 0 < 1, and 1 ~< p < ec, Xo,p denotes <strong>the</strong> space<br />

/C(X, ~p, a, b) where a2n "- e On, bzn "- e -(1-0)n, azn+l "- e -On, b2n+l "- e (1-0)n. In-<br />

stead <strong>of</strong> us<strong>in</strong>g [I 9 II/c(N,a,b~ on Xo,p, it is customary to use<br />

Ilxll0,p --<strong>in</strong>f [le~ v II e- -~<br />

where <strong>the</strong> <strong>in</strong>fimum is over all x0(t), Xl (t) for which e ~ xo(t) E Lp(R, X0), e-(1-~ (t) c<br />

Lp(R, X1), and x - xo(t) + xl (t) for every t <strong>in</strong> R. The expression II 9 I]0,p is a norm on<br />

Xo,p which is equivalent to II 9 II~c~,a,b> but is better behaved. For example, (Xo,p, I1" II0,p)

80 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

is uniformly convex (ra<strong>the</strong>r than just superreflexive) if ei<strong>the</strong>r X0 or X 1 is uniformly convex<br />

and 1 < p < cx~ [15, 2.g.21]. Moreover, I]" IIO,p is very good for <strong>in</strong>terpolation purposes,<br />

for if Y is ano<strong>the</strong>r <strong>Banach</strong> couple <strong>the</strong>n (Xo,p, ]]. ]]0,p) and (Yo,p, ][" ]]0,p) are an exact<br />

<strong>in</strong>terpolation pair <strong>of</strong> exponent 0 with respect to X and Y.<br />

It is <strong>of</strong> course important to identify Xo,p when X is a concrete <strong>Banach</strong> couple. The<br />

spaces that arise <strong>in</strong> this connection when X is a couple <strong>of</strong> L p(#) spaces are <strong>the</strong> L p,q<br />

spaces. Given 0 < p < ~, 0 < q < cx~, a measure #, and a <strong>Banach</strong> space X, Lp,q (#, X)<br />

is <strong>the</strong> space <strong>of</strong> X valued strongly measurable functions x for which<br />

I]Xllpq "-- q/p [tl/px.(t)]q dt 1/q < o~, (42)<br />

( f0 t )<br />

where x*(t) is <strong>the</strong> decreas<strong>in</strong>g rearrangement <strong>of</strong> IIx(t)llx. For 0 < p ~< cx~, Lp,~(#, X) is<br />

<strong>the</strong> space <strong>of</strong> X valued strongly measurable functions for which<br />

IlXllp,~ "-suptl/Px*(t) < ~. (43)<br />

t>0<br />

When X is <strong>the</strong> scalar field we write Lp,q(#). Evidently Lp,p(#, X) = Lp(#, X). Note<br />

that for p > q ~> 1, <strong>the</strong> space Lp,q(#) is <strong>the</strong> Lorentz function space Lw, q(#) def<strong>in</strong>ed<br />

<strong>in</strong> Section 5 with W(t) --qt q/p-I and hence is a <strong>Banach</strong> space. When q > p ~> 1 <strong>the</strong><br />

expression II 9 I] p,q does not satisfy <strong>the</strong> triangle <strong>in</strong>equality, although II 9 Ilp,q is equivalent to<br />

a norm when p > 1. In any case Lp,q (#, X) is a metrizable topological vector space.<br />

The ma<strong>in</strong> result about spaces obta<strong>in</strong>ed from L p and L p,q spaces via <strong>the</strong>/C-method is<br />

<strong>the</strong> follow<strong>in</strong>g (see [4, 5.3.1]): Let 0 < Po, Pl, qo, ql 0, C-1]] 9 ]]p,q ~ ]] 9 ][O,q<br />

C II 9 II p,q. When <strong>the</strong> expressions II 9 II Pi ,qi' i -- 0, 1, are equivalent to norms one can deduce<br />

from (44) and earlier comments an <strong>in</strong>terpolation <strong>the</strong>orem for operators. In fact, <strong>the</strong>re are<br />

cases where <strong>in</strong>terpolation is valid even when <strong>the</strong> spaces are not all <strong>Banach</strong> spaces. In particular,<br />

assume that T" Lpi,r i (l~, X) -+ Lqi,s i (19, X) is cont<strong>in</strong>uous for i -- 0, 1, with P0 ~: Pl,<br />

q0 ~ ql, 0 < 0 < 1, and def<strong>in</strong>e p, q by 1/p - (1 -O)/po + O/pl, 1/q - (1 -O)/qo --<br />

O/ql. If p ~< q, <strong>the</strong>n T'Lp(lZ, X) ~ Lq(19, Y) is cont<strong>in</strong>uous and for 0

<strong>Basic</strong> concepts <strong>in</strong> <strong>the</strong> geometry <strong>of</strong> <strong>Banach</strong> spaces 81<br />

that X is obta<strong>in</strong>able from X via <strong>the</strong>/C-method. This <strong>the</strong>orem says <strong>the</strong> follow<strong>in</strong>g. If X is ~Cmonotone<br />

for X, <strong>the</strong>n <strong>the</strong>re are E, a, and b so that X =/C(X, E, a, b), up to an equivalent<br />

renorm<strong>in</strong>g.<br />

The pair (Lp(O, 1), Lq(O, 1)), 1 ~< p < q ~< ~, is a Calder6n couple, and many o<strong>the</strong>r<br />

examples are known (see [31]). One <strong>in</strong>terest<strong>in</strong>g problem which is not completely solved<br />

is to determ<strong>in</strong>e a necessary and sufficient condition for a <strong>Banach</strong> couple <strong>of</strong> symmetric lattice<br />

ideals on [0, 1] (see Section 5) to be a Calder6n couple. Much is known about <strong>the</strong><br />

<strong>in</strong>terpolation spaces for such a couple. For example, an exact <strong>in</strong>terpolation space with<br />

respect to such a couple must itself be a symmetric lattice ideal (<strong>the</strong> first step is to observe<br />

that if r'[0, 1] --+ [0, 1] is a measure preserv<strong>in</strong>g automorphism, <strong>the</strong>n x w-> x(r)<br />

def<strong>in</strong>es an isometric automorphism <strong>of</strong> any symmetric lattice on [0, 1]). When <strong>the</strong> couple<br />

is (L1 (0, 1), L~(0, 1)), it is reasonable to guess that <strong>the</strong> converse is true. It is not,<br />

but: If L~(O, 1) is dense <strong>in</strong> <strong>the</strong> symmetric lattice ideal X, <strong>the</strong>n X is an exact <strong>in</strong>terpolation<br />

space with respect to (Ll (0, 1), L~(0, 1)) [13, Theorem 4.10]. Most natural symmetric<br />

lattice ideals on [0, 1], <strong>in</strong>clud<strong>in</strong>g all <strong>the</strong> separable ones, satisfy <strong>the</strong> hypo<strong>the</strong>sis <strong>of</strong> this<br />

<strong>the</strong>orem.<br />

12. List <strong>of</strong> symbols<br />

Here is a list <strong>of</strong> symbols used <strong>in</strong> this <strong>in</strong>troductory article and, where appropriate, a reference<br />

to where <strong>the</strong>y are def<strong>in</strong>ed.<br />

N<br />

R<br />

C<br />

T<br />

IP<br />

E<br />

Ap, Bp<br />

KG<br />

C(K;X)<br />

C(K)<br />

L p (l,t, X)<br />

Lp(#)<br />

L~(#,X)<br />

Lp(O, 1)<br />

Cp(~r)<br />

ep(r)<br />

The natural numbers.<br />

The real numbers.<br />

The complex numbers.<br />

The complement <strong>of</strong> <strong>the</strong> set S.<br />

The unit circle <strong>in</strong> <strong>the</strong> complex plane.<br />

A probability measure (Section 2).<br />

f. dip (Section 2).<br />

The constants <strong>in</strong> Kh<strong>in</strong>tch<strong>in</strong>e's <strong>in</strong>equality (1); also <strong>the</strong> constants <strong>in</strong> <strong>the</strong><br />

Kahane-Kh<strong>in</strong>tch<strong>in</strong>e <strong>in</strong>equality (24).<br />

The constant <strong>in</strong> Gro<strong>the</strong>ndieck's <strong>in</strong>equality (Section 10).<br />

Cont<strong>in</strong>uous functions f on <strong>the</strong> (usually) compact Hausdorff space K<br />

tak<strong>in</strong>g values <strong>in</strong> <strong>the</strong> (usually) normed space X, normed by ]lfll =<br />

suPtcK II f(t) II.<br />

C (K; X) when X is <strong>the</strong> scalar field.<br />

The #-measurable X-valued functions f for which<br />

Ilfllp := (f IIf[I F d#) 1/p < ~ (Section 7). Here 0 < p < ec.<br />

L p (#, X) when X = R.<br />

The #-measurable essentially bounded X-valued functions, with norm<br />

11 f II ~ := <strong>in</strong>f~A=O sup I fl A I"<br />

L p (#) when # is Lebesgue measure on <strong>the</strong> unit <strong>in</strong>terval.<br />

Lp(#) when # is normalized Lebesgue measure on <strong>the</strong> unit circle.<br />

Lp(#) when g is count<strong>in</strong>g measure on <strong>the</strong> set F.

82 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

~.p<br />

17 ~p<br />

C<br />

co(F)<br />

CO<br />

L p,q (/z, X)<br />

L p, oc (/z , X)<br />

L p,q (/z )<br />

Bx<br />

Bx(x,r)<br />

s-L<br />

S_L<br />

d(X, Y)<br />

X ,~ Y<br />

~x(.)<br />

px(.)<br />

Xu<br />

Xu,<br />

Ix<br />

Jx<br />

Ip<br />

Ip,q<br />

B(X,Y)<br />

K(X, Y)<br />

WK(X, Y)<br />

SS(X, Y)<br />

Fr(X, Y)<br />

.N'(X, Y)<br />

A/'(T)<br />

M(P)(T)<br />

M(p)(T)<br />

Tp(X)<br />

Cp(X)<br />

g~p(F) when F = N.<br />

g~p(F) when F -- { 1, 2 ..... n}.<br />

The subspace <strong>of</strong> s <strong>of</strong> scalar sequences which have a limit.<br />

The closure <strong>in</strong> s (F) <strong>of</strong> <strong>the</strong> scalar sequences which have f<strong>in</strong>ite support.<br />

co (F) when F -- N.<br />

The /z-measurable X-valued functions f for which ][f][p,q "=<br />

(q fo ][tl/pf *(t)][q ~)l/q < co(Section 11). Here 0 < p,q < co and<br />

f* is <strong>the</strong> decreas<strong>in</strong>g rearrangement <strong>of</strong> []fllx.<br />

The /z-measurable X-valued functions f for which [1 f ]]p,oc "=<br />

supt>otl/Pl[f*(t)l[x < co (Section 11). Here 0 < p ~< co and f* is<br />

<strong>the</strong> decreas<strong>in</strong>g rearrangement <strong>of</strong> 11 f ]Ix.<br />

Lp,q (/z, X) when X -- ]R.<br />

The closed unit ball <strong>of</strong> <strong>the</strong> <strong>Banach</strong> space X.<br />

The closed ball <strong>of</strong> radius r with center x <strong>in</strong> <strong>the</strong> <strong>Banach</strong> space X; denoted<br />

also B(x, r) when X is understood.<br />

All l<strong>in</strong>ear functionals which vanish on S (when S is a subset <strong>of</strong> a <strong>Banach</strong><br />

space).<br />

The <strong>in</strong>tersection <strong>of</strong> <strong>the</strong> kernels <strong>of</strong> all l<strong>in</strong>ear functionals <strong>in</strong> S (when S is<br />

a subset <strong>of</strong> <strong>the</strong> dual <strong>of</strong> a <strong>Banach</strong> space).<br />

The <strong>Banach</strong>-Mazur distance from X to Y (Section 2).<br />

The space X is isomorphic to <strong>the</strong> space Y.<br />

The modulus <strong>of</strong> convexity <strong>of</strong> <strong>the</strong> space X (Section 6).<br />

The modulus <strong>of</strong> smoothness <strong>of</strong> <strong>the</strong> space X (Section 6).<br />

When X is a lattice and u ~> 0, <strong>the</strong> abstract M-space which has <strong>the</strong> order<br />

<strong>in</strong>terval [-u, u] as <strong>the</strong> unit ball (Section 5).<br />

When X is a lattice and u* ~> 0 <strong>in</strong> X*, <strong>the</strong> abstract L 1-space which is<br />

<strong>the</strong> completion <strong>of</strong> X under <strong>the</strong> sem<strong>in</strong>orm [[x ]]u. -- u* ([x[) (Section 5).<br />

A Rademacher sequence (Section 4).<br />

The identity operator on <strong>the</strong> space X.<br />

The canonical embedd<strong>in</strong>g <strong>of</strong> X <strong>in</strong>to X**.<br />

The formal identity operator from C(K) to Lp(/z) (when/z is a f<strong>in</strong>ite<br />

measure on <strong>the</strong> compact Hausdorff space K).<br />

The formal identity mapp<strong>in</strong>g from Lp(/z) to Lq(/z).<br />

The bounded operators from X to Y.<br />

The compact operators from X to Y.<br />

The weakly compact operators from X to Y.<br />

The strictly s<strong>in</strong>gular operators from X to Y (Section 10).<br />

The Fredholm operators from X to Y (Section 10).<br />

The nuclear operators from X to Y (Section 8).<br />

The nuclear norm <strong>of</strong> <strong>the</strong> operator T (Section 8).<br />

The p-convexity constant <strong>of</strong> <strong>the</strong> operator T (Section 5).<br />

The p-concavity constant <strong>of</strong> <strong>the</strong> operator T (Section 5).<br />

The type p constant <strong>of</strong> <strong>the</strong> <strong>Banach</strong> space X (Section 8).<br />

The cotype p constant <strong>of</strong> <strong>the</strong> <strong>Banach</strong> space X (Section 8).

83<br />

ITp(X, Y)<br />

7rp(T)<br />

In(X, Y)<br />

ip(T)<br />

The p-summ<strong>in</strong>g operators from X to Y (Section 10).<br />

The p-summ<strong>in</strong>g norm <strong>of</strong> <strong>the</strong> operator T (Section 10).<br />

The p-<strong>in</strong>tegral operators from X to Y (Section 10).<br />

The p-<strong>in</strong>tegral norm <strong>of</strong> <strong>the</strong> operator T (Section 10).<br />

References<br />

[1] C.D. Aliprantis and O. Burk<strong>in</strong>shaw, Positive Operators, Academic Press, New York, 1985.<br />

[2] B. Beauzamy, Introduction to <strong>Banach</strong> <strong>Spaces</strong> and Their <strong>Geometry</strong>, Ma<strong>the</strong>matics Studies 68, North-Holland,<br />

Amsterdam (1985).<br />

[3] Y. Benyam<strong>in</strong>i and J. L<strong>in</strong>denstrauss, Geometric Nonl<strong>in</strong>ear Functional Analysis, Coll. Pub. 48, Amer. Math.<br />

Soc., Providence, RI (2000).<br />

[4] J. Bergh and J. Lrfstrrm, Interpolation <strong>Spaces</strong>: An Introduction, Grundlehren der Ma<strong>the</strong>matischen Wissenschaften<br />

223, Spr<strong>in</strong>ger-Verlag, New York (1976).<br />

[5] Yu.A. Brudnyi and N.Ya. Kruglyak, Interpolation Functors and Interpolation <strong>Spaces</strong> I, North-Holland,<br />

Amsterdam (1991).<br />

[6] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renorm<strong>in</strong>gs <strong>in</strong> <strong>Banach</strong> <strong>Spaces</strong>, Longman Scientific<br />

& Technical, Essex (1993).<br />

[7] J. Diestel, Sequences and Series <strong>in</strong> <strong>Banach</strong> <strong>Spaces</strong>, Spr<strong>in</strong>ger-Verlag, New York (1984).<br />

[8] J. Diestel and J.J. Uhl, Vector Measures, Ma<strong>the</strong>matical Surveys 15, Amer. Math. Soc., Providence, RI<br />

(1977).<br />

[9] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summ<strong>in</strong>g Operators, Cambridge Studies <strong>in</strong> Advanced<br />

Math. 43, Cambridge University Press, Cambridge (1995).<br />

[10] R. Durrett, Probability: Theory and Examples, Duxbury Press (1996).<br />

[ 11 ] P. Habala, P. H~ijek and V. Zizler, Introduction to <strong>Banach</strong> <strong>Spaces</strong> I, Matfyzpress, Univerzity Karlovy (1996).<br />

[ 12] P. Habala, P. Hfijek and V. Zizler, Introduction to <strong>Banach</strong> <strong>Spaces</strong> II, Matfyzpress, Univerzity Karlovy (1996).<br />

[13] S.G. Kre<strong>in</strong>, Yu.I. Petun<strong>in</strong> and E.M. Semenov, Interpolation <strong>of</strong> L<strong>in</strong>ear Operators, Transl. Math. Monogr. 54,<br />

Amer. Math. Soc., Providence, RI (1982).<br />

[14] J. L<strong>in</strong>denstrauss and L. Tzafriri, Classical <strong>Banach</strong> <strong>Spaces</strong> I: Sequence <strong>Spaces</strong>, Ergebnisse der Ma<strong>the</strong>matik<br />

und ihrer Grenzgebiete 92, Spr<strong>in</strong>ger-Verlag, Berl<strong>in</strong> (1977).<br />

[15] J. L<strong>in</strong>denstrauss and L. Tzafriri, Classical <strong>Banach</strong> <strong>Spaces</strong> II: Function <strong>Spaces</strong>, Ergebnisse der Ma<strong>the</strong>matik<br />

und ihrer Grenzgebiete 97, Spr<strong>in</strong>ger-Verlag, Berl<strong>in</strong> (1979).<br />

[16] V.D. Milman and G. Schechtman, Asymptotic Theory <strong>of</strong> F<strong>in</strong>ite Dimensional Normed <strong>Spaces</strong>, Lecture Notes<br />

<strong>in</strong> Math. 1200, Spr<strong>in</strong>ger-Verlag, Berl<strong>in</strong> (1986).<br />

[17] G. Pisier, The Volume <strong>of</strong> Convex Bodies and <strong>Banach</strong> Space <strong>Geometry</strong>, Cambridge Tracts <strong>in</strong> Math. 94,<br />

Cambridge University Press, Cambridge (1989).<br />

[18] H.L. Royden, Real Analysis, 3rd edn., Macmillan, New York (1988).<br />

[19] W. Rud<strong>in</strong>, Functional Analysis, 2nd edn., McGraw-Hill, New York (1991).<br />

[20] N. Tomczak-Jaegermann, <strong>Banach</strong>-Mazur Distances and F<strong>in</strong>ite-Dimensional Operator Ideals, Longman<br />

Scientific & Technical, Essex (1989).<br />

[21 ] P. Wojtaszczyk, <strong>Banach</strong> <strong>Spaces</strong> for Analysts, Cambridge Studies <strong>in</strong> Advanced Ma<strong>the</strong>matics 25, Cambridge<br />

University Press, Cambridge (1991).<br />

Handbook articles which are referenced <strong>in</strong> this <strong>in</strong>troductory article<br />

[22] D. Alspach and E. Odell, L p spaces, This Handbook.<br />

[23] D.L. Burkholder, Mart<strong>in</strong>gales and s<strong>in</strong>gular <strong>in</strong>tegrals <strong>in</strong> <strong>Banach</strong> spaces, This Handbook.<br />

[24] EG. Casazza, Approximation properties, This Handbook.<br />

[25] R. Deville and N. Ghoussoub, Perturbed m<strong>in</strong>imization pr<strong>in</strong>ciples and applications, This Handbook.<br />

[26] V. Fonf, J. L<strong>in</strong>denstrauss and R.R. Phelps, Inf<strong>in</strong>ite dimensional convexity, This Handbook.

84 W.B. Johnson and J. L<strong>in</strong>denstrauss<br />

[27]<br />

[281<br />

[29]<br />

[30]<br />

[31]<br />

[32]<br />

[33]<br />

[34]<br />

[35]<br />

[36]<br />

[37]<br />

[381<br />

[39]<br />

[401<br />

[41]<br />

[421<br />

[43]<br />

[441<br />

T. Gamel<strong>in</strong> and S.V. Kislyakov, Uniform algebras as <strong>Banach</strong> spaces, This Handbook.<br />

A.A. Giannopoulos and V. Milman, Euclidean structure <strong>in</strong> f<strong>in</strong>ite dimensional normed spaces, This Handbook.<br />

G. Godefroy, Renorm<strong>in</strong>gs <strong>of</strong> <strong>Banach</strong> spaces, This Handbook.<br />

W.T. Gowers, Ramsey <strong>the</strong>ory methods <strong>in</strong> <strong>Banach</strong> spaces, This Handbook.<br />

N.J. Kalton and S.J. Montgomery-Smith, Interpolation andfactorization <strong>the</strong>orems, This Handbook.<br />

S.V. Kislyakov, <strong>Banach</strong> spaces and classical harmonic analysis, This Handbook.<br />

A. Koldobsky and H. K6nig, Aspects <strong>of</strong> <strong>the</strong> isometric <strong>the</strong>ory <strong>of</strong> <strong>Banach</strong> spaces, This Handbook.<br />

M. Ledoux and J. Z<strong>in</strong>n, Probabilistic limit <strong>the</strong>orems <strong>in</strong> <strong>the</strong> sett<strong>in</strong>g <strong>of</strong> <strong>Banach</strong> spaces, This Handbook.<br />

J. L<strong>in</strong>denstrauss, Characterizations <strong>of</strong> Hilbert space, This Handbook.<br />

E Mankiewicz and N. Tomczak-Jaegermann, Quotients <strong>of</strong>f<strong>in</strong>ite-dimensional <strong>Banach</strong> spaces; Random phenomena,<br />

This Handbook.<br />

B. Maurey, <strong>Banach</strong> spaces with few operators, This Handbook.<br />

B. Maurey, Type, cotype and K-convexity, This Handbook.<br />

D. Preiss, Geometric measure <strong>the</strong>ory <strong>in</strong> <strong>Banach</strong> spaces, This Handbook.<br />

H.E Rosenthal, The <strong>Banach</strong> spaces C (K), This Handbook.<br />

L. Tzafriri, Uniqueness <strong>of</strong> structure <strong>in</strong> <strong>Banach</strong> spaces, This Handbook.<br />

P. Wojtaszczyk, <strong>Spaces</strong> <strong>of</strong> analytic functions with <strong>in</strong>tegral norm, This Handbook.<br />

M. Zipp<strong>in</strong>, Extension <strong>of</strong> bounded l<strong>in</strong>ear operators, This Handbook.<br />

V. Zizler, Nonseparable <strong>Banach</strong> spaces, This Handbook.

CHAPTER 2<br />

Positive Operators<br />

Y.A. Abramovich<br />

Department <strong>of</strong> Ma<strong>the</strong>matical Sciences, Indiana University-Purdue University, Indianapolis, IN, USA<br />

E-mail: yabramovich @ math. iupui, edu<br />

C.D. Aliprantis<br />

Department <strong>of</strong> Economics and Department <strong>of</strong> Ma<strong>the</strong>matics, Purdue University, West Lafayette, IN, USA<br />

E-mail: aliprantis @ mgmt.purdue, edu<br />

Contents<br />

1. Ordered vector spaces and <strong>Banach</strong> lattices .................................. 87<br />

2. Operators between <strong>Banach</strong> lattices ...................................... 90<br />

3. When is every cont<strong>in</strong>uous operator regular? ................................. 91<br />

4. Dom<strong>in</strong>ation, compactness, and factorization ................................. 93<br />

5. Invariant subspaces <strong>of</strong> positive operators ................................... 98<br />

6. Compact-friendly operators and <strong>in</strong>variant subspaces ............................. 102<br />

7. Integral operators and <strong>in</strong>variant subspaces .................................. 105<br />

8. Applications ................................................... 111<br />

Acknowledgment .................................................. 117<br />

Added <strong>in</strong> Pro<strong>of</strong> ................................................... 117<br />

References ..................................................... 117<br />

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 1<br />

Edited by William B. Johnson and Joram L<strong>in</strong>denstrauss<br />

9 2001 Elsevier Science B.V. All fights reserved<br />

85

Positive operators 87<br />

The <strong>the</strong>ory <strong>of</strong> positive operators is a dist<strong>in</strong>guished and significant part <strong>of</strong> <strong>the</strong> field <strong>of</strong><br />

general operator <strong>the</strong>ory. The extra feature <strong>of</strong> this part <strong>of</strong> operator <strong>the</strong>ory is <strong>the</strong> existence<br />

<strong>of</strong> an order on <strong>the</strong> spaces <strong>in</strong>volved. This <strong>in</strong>gredient is <strong>of</strong> tremendous importance and lies<br />

<strong>in</strong> <strong>the</strong> core <strong>of</strong> many specific results valid for positive operators. The two ma<strong>in</strong> objectives<br />

<strong>of</strong> <strong>the</strong> present work are to discuss <strong>the</strong> relationships between general operators and positive<br />

operators and to demonstrate <strong>the</strong> effects <strong>the</strong> order structure has on general operators act<strong>in</strong>g<br />

between <strong>Banach</strong> lattices.<br />

We list here several books devoted primarily to positive operators between ordered<br />

spaces: [15,22,29,41,75,78-80,86,87,96,100,123,126,135,144,145]. The size constra<strong>in</strong>ts<br />

<strong>of</strong> this article as well as <strong>the</strong> areas <strong>of</strong> <strong>in</strong>terest <strong>of</strong> <strong>the</strong> authors have <strong>in</strong>evitably <strong>in</strong>fluenced<br />

<strong>the</strong> selection <strong>of</strong> <strong>the</strong> material for this survey and have precluded us from mention<strong>in</strong>g many<br />

exist<strong>in</strong>g directions with<strong>in</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> positive operators. We especially regret hav<strong>in</strong>g to<br />

omit <strong>the</strong> ergodic and <strong>in</strong>terpolation properties <strong>of</strong> positive operators. For <strong>the</strong>se topics <strong>the</strong><br />

reader is referred to [83] and [84] respectively. S<strong>in</strong>ce our survey is devoted to operators,<br />

we have reduced to a bare m<strong>in</strong>imum <strong>the</strong> general <strong>the</strong>ory <strong>of</strong> <strong>Banach</strong> lattices presented here.<br />

Most <strong>of</strong> <strong>the</strong> books cited above, <strong>the</strong> series <strong>of</strong> notes by Luxemburg and Zaanen [93,92] and<br />

<strong>the</strong> two surveys [39,40] are excellent sources for such a <strong>the</strong>ory as well. Throughout this<br />

work <strong>the</strong> word "operator" is synonymous with "l<strong>in</strong>ear operator".<br />

1. Ordered vector spaces and <strong>Banach</strong> lattices<br />

Though we will be deal<strong>in</strong>g mostly with operators on <strong>Banach</strong> lattices, it is fruitful to start<br />

with <strong>the</strong> general framework <strong>of</strong> (partially) ordered vector spaces. We will <strong>in</strong>troduce some<br />

necessary term<strong>in</strong>ology and prove a few basic results; for a systematic presentation we refer<br />

<strong>the</strong> reader to [70,94,101,107,134,143].<br />

Recall that a real vector space X equipped with a partial order ~> is said to be a (partially)<br />

ordered vector space whenever x ~ y imply otx ~> oty for all ol ~> 0 and x + z ~> y + z for<br />

all z E X. The set X+ = {x E X: x ~> 0} is called <strong>the</strong> positive cone <strong>of</strong> X and its elements<br />

are referred to as positive vectors. The cone X+ is said to be generat<strong>in</strong>g whenever X --<br />

X+ - X+, i.e., whenever every vector can be written as a difference <strong>of</strong> two positive vectors.<br />

An ordered vector space X is said to be Archimedean whenever nx 0 or x ~< 0. E]<br />

THEOREM 2 (M. Kre<strong>in</strong>-Smulian). Let X be a <strong>Banach</strong> space ordered by a closed generat<strong>in</strong>g<br />

cone. Then <strong>the</strong>re is a constant M > 0 such that for each x ~ X <strong>the</strong>re are x l, x2 E X+<br />

satisfy<strong>in</strong>g x -- xl - x2 and Ilxi II ~< Mllxll for each i.

88 Y.A. Abramovich and C.D. Aliprantis<br />

PROOF. We present a sketch <strong>of</strong> <strong>the</strong> pro<strong>of</strong>. For each n def<strong>in</strong>e <strong>the</strong> set<br />

En = {x ~ X" =tXl, X2 G_ X- t- with x - Xl - x2 and [Ixi II ~ n (i = 1, 2) }.<br />

Clearly, each En is convex, symmetric, and 0 6 En. In addition, note that En cc_ Em whenever<br />

n ~< m.<br />

S<strong>in</strong>ce X+ is generat<strong>in</strong>g, we see that X - Un~=l En. So, by <strong>the</strong> Baire Category Theorem,<br />

some Ek conta<strong>in</strong>s a closed ball B(xo, r) = {x 6 X: Ilx0-xll ~< r}. The properties <strong>of</strong><strong>the</strong> sets<br />

En imply that B(0, r) c_ E~. Now by imitat<strong>in</strong>g <strong>the</strong> pro<strong>of</strong> <strong>of</strong> <strong>the</strong> open mapp<strong>in</strong>g <strong>the</strong>orem, we<br />

can show that B(O, r) c_ Ezk holds and <strong>the</strong> pro<strong>of</strong> is done. The details can be found <strong>in</strong> [135];<br />

see also [4].<br />

D<br />

COROLLARY 3. Let X be an ordered <strong>Banach</strong> space whose positive cone is closed and<br />

generat<strong>in</strong>g. If xn --+ x holds <strong>in</strong> X, <strong>the</strong>n <strong>the</strong>re exist Yn, Zn, Y, z ~ X+ such that Xn -- Yn -- Zn,<br />

x = y -- Z, Yn --+ Y and Zn --+ z.<br />

COROLLARY 4. Let X be a <strong>Banach</strong> space partially ordered by a closed generat<strong>in</strong>g cone,<br />

and let Y be a topological vector space. Then an operator T : X --+ Y is cont<strong>in</strong>uous if and<br />

only if <strong>the</strong> restriction <strong>of</strong> T to X+ is cont<strong>in</strong>uous.<br />

And now we <strong>in</strong>troduce <strong>the</strong> central concept <strong>of</strong> this work.<br />

DEFINITION 5. An operator T:X ~ Y between two ordered vector spaces is said to be<br />

positive if T (X+) _ Y+, i.e., if x ~> 0 implies Tx >~ O.<br />

It is a remarkable fact that quite <strong>of</strong>ten positive operators are automatically cont<strong>in</strong>uous.<br />

This was first proven by M. Kre<strong>in</strong> for positive l<strong>in</strong>ear functionals [81 ] and later was generalized<br />

<strong>in</strong> several contexts by various authors; see, for <strong>in</strong>stance, [32,99,101,121]. The next<br />

result, due to Lozanovsky, is <strong>the</strong> strongest <strong>in</strong> this direction and appeared <strong>in</strong> [136].<br />

COROLLARY 6 (Lozanovsky). Let X and Y be two ordered <strong>Banach</strong> spaces whose cones<br />

are closed. If additionally, <strong>the</strong> cone <strong>of</strong> X is generat<strong>in</strong>g, <strong>the</strong>n every positive operator<br />

T : X --+ Y is cont<strong>in</strong>uous.<br />

PROOF. It suffices to show that a positive operator T has a closed graph. So, assume<br />

Xn --+ 0 <strong>in</strong> X and Txn --+ y <strong>in</strong> Y. By pass<strong>in</strong>g to a subsequence, we can also assume<br />

oo<br />

Y~n=l n ]lXn I] < cx~. By Theorem 2 <strong>the</strong>re exist some M > 0 and two sequences {Yn } and<br />

{Zn} <strong>of</strong> X+ satisfy<strong>in</strong>g Xn = Yn - Zn, Ilynl] ~< m[Ixnll, and I[Zn [] ~< mllxnl[ for each n. S<strong>in</strong>ce<br />

X+ is closed, <strong>the</strong> vector z = ~-~n~=l n(yn + Zn) <strong>of</strong> X belongs to X+ and -z

Positive operators 89<br />

A norm I1" II on a vector lattice is said to be a lattice norm if Ixl ~ lyl implies Ilxll ~ Ilyll.<br />

A normed vector lattice is a vector lattice equipped with a lattice norm. A norm complete<br />

normed vector lattice is called a <strong>Banach</strong> lattice. Here is a list <strong>of</strong> <strong>the</strong> most important proper-<br />

ties between <strong>the</strong> topological and order structures <strong>of</strong> a <strong>Banach</strong> lattice that will be employed<br />

<strong>in</strong> this work. The first one, <strong>the</strong> order cont<strong>in</strong>uity, is discussed at length <strong>in</strong> [72].<br />

DEFINITION 7. A lattice norm on a vector lattice X is said to be:<br />

(1) order cont<strong>in</strong>uous, if xc~ $ 0 implies ]]x~ 11 $ 0;<br />

(2) a Levi norm, if 0 ~< x~ 1" and ]lx~ 11 ~< 1 imply that sup~ x~ exists <strong>in</strong> X;<br />

(3) a Fatou norm, if 0 ~< xc~ 1" x implies Ilx~ II 1" Ilxll;<br />

(4) a weak Fatou norm, if <strong>the</strong>re exists some constant K ~> 1 such that<br />

0 ~ xo~ 1" x ------->, IIx II ~ K lim Ilxo~ II.<br />

Of<br />

A <strong>Banach</strong> lattice with an order cont<strong>in</strong>uous Levi norm is usually referred to as a<br />

Kantorovich-<strong>Banach</strong> space, or as a KB-space <strong>in</strong> short. The mean<strong>in</strong>gs <strong>of</strong> <strong>the</strong> expressions<br />

"X is a Levi <strong>Banach</strong> lattice", or "X has <strong>the</strong> Levi property", or "X has <strong>the</strong> weak Fatou<br />

property", etc. should be clear and <strong>the</strong>y will be used throughout this work.<br />

If X has a Levi norm, <strong>the</strong>n obviously X is Dedek<strong>in</strong>d complete. Note that two equivalent<br />

(lattice) norms are ei<strong>the</strong>r both Levi or both fail to be Levi. The same is true for <strong>the</strong> weak<br />

Fatou property. It is well known [30] that if a norm is Levi, <strong>the</strong>n necessarily it is also weak<br />

Fatou. Property (3) is clearly an isometric property.<br />

If, <strong>in</strong> <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> an order cont<strong>in</strong>uous, or a Levi, or a (weak) Fatou norm one replaces<br />

arbitrary nets by sequences, <strong>the</strong>n one arrives at <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> a sequentially order<br />

cont<strong>in</strong>uous norm, or a sequentially Levi norm, or respectively, <strong>of</strong> a sequentially (weak)<br />

Fatou norm. 1 Thus, for example, a norm on a <strong>Banach</strong> lattice X is sequentially Fatou (or<br />

simply ~-Fatou) if for each sequence 0 ~< xn t x we have ]Ix ]1 = lim ]]xn 11.<br />

There is a very rich structural <strong>the</strong>ory <strong>of</strong> <strong>Banach</strong> lattices (reflect<strong>in</strong>g partly <strong>the</strong> correspond-<br />

<strong>in</strong>g <strong>the</strong>ory <strong>of</strong> <strong>Banach</strong> spaces) <strong>in</strong> which various properties <strong>of</strong> <strong>Banach</strong> lattices are character-<br />

ized by <strong>the</strong> absence or presence <strong>of</strong> subspaces or vector sublattices which are isomorphic to<br />

some classical spaces ~p (1 ~< p ~< ec) or co. We refer to [29,39,40,96] for <strong>the</strong>se results.<br />

As an illustration, we mention only <strong>the</strong> famous <strong>the</strong>orem stat<strong>in</strong>g that a <strong>Banach</strong> lattice X is<br />

a KB-space if and only if X does not conta<strong>in</strong> a vector sublattice order isomorphic to co.<br />

1 These properties appear <strong>in</strong> <strong>the</strong> literature under many different names. Freml<strong>in</strong> [58] was <strong>the</strong> first to associate <strong>the</strong><br />

Levi and Fatou names with <strong>the</strong>se properties. Nakano [100, pp. 129-130] used <strong>the</strong> term monotone complete norm<br />

for a sequentially Levi norm and universally monotone complete norm for a Levi norm. Zaanen [144] considers<br />

sequentially Levi norms <strong>in</strong> two places under different names. First on p. 305 he refers to <strong>the</strong>m, like Nakano, as<br />

monotone complete norms, and <strong>the</strong>n on p. 421 as norms with <strong>the</strong> weak Fatou property for monotone sequences.<br />

The term weak Fatou property for directed sets (p. 390) is used by Zaanen for what we refer to as a Levi norm.<br />

Meyer-Nieberg [96, p. 96], uses <strong>the</strong> term "monotonically complete norm" for a Levi norm. The Soviet school on<br />

<strong>Banach</strong> lattices used <strong>the</strong> symbols (A) and (A f) to denote <strong>the</strong> sequential order cont<strong>in</strong>uous and order cont<strong>in</strong>uous<br />

norms, respectively; <strong>the</strong>y also used <strong>the</strong> symbols (B) and (B r) to denote <strong>the</strong> sequential Levi and <strong>the</strong> Levi properties<br />

respectively, and f<strong>in</strong>ally <strong>the</strong>y used <strong>the</strong> symbols (C) and (C f) to denote <strong>the</strong> sequential Fatou and <strong>the</strong> Fatou properties<br />

respectively.

90 YA. Abramovich and C.D. Aliprantis<br />

A vector subspace J <strong>of</strong> a vector lattice X is said to be an (order) ideal if Ixl ~ lyl and<br />

y 6 J imply that x 6 J. For each 0 0 such that Ix[ ~< )~u}.<br />

A positive element u 6 X is called a (strong) unit if Xu = X. Each unit generates <strong>the</strong> so<br />

called unit-norm II" Ilu via IIxllu = <strong>in</strong>f{)~ >~ 0: Ix[ ~< ~.u}. Clearly this norm is always Fatou;<br />

if X is a-Dedek<strong>in</strong>d complete, <strong>the</strong>n I]" Ilu is complete.<br />

2. Operators between <strong>Banach</strong> lattices<br />

If X and Y are <strong>Banach</strong> spaces, <strong>the</strong>n <strong>the</strong> symbol s Y) denotes <strong>the</strong> <strong>Banach</strong> space <strong>of</strong> all<br />

cont<strong>in</strong>uous operators from X to Y; we let s = E(X, X). 2 If X and Y are <strong>in</strong> addition<br />

<strong>Banach</strong> lattices, <strong>the</strong>n <strong>the</strong>re are several very important classes <strong>of</strong> operators associated with<br />

<strong>the</strong> order structure on <strong>the</strong>se spaces. We denote by 12+ (X, Y) <strong>the</strong> collection <strong>of</strong> all positive<br />

operators. This collection is a cone and it <strong>in</strong>duces a natural (partial) order on <strong>the</strong> <strong>Banach</strong><br />

space C(X, Y).<br />

The subspace s Y) - E+(X, Y) <strong>of</strong>/2(X, Y) generated by <strong>the</strong> cone/2+(X, Y) is<br />

denoted by/2 r (X, Y), or simply/2 r, and is referred to as <strong>the</strong> space <strong>of</strong> regular operators.<br />

In o<strong>the</strong>r words, an operator is said to be regular if it can be written as a difference <strong>of</strong> two<br />

positive operators.<br />

An operator T" X --+ Y is order bounded if T maps order bounded sets <strong>in</strong> X to order<br />

bounded sets <strong>in</strong> Y. The symbol/2 b (X, Y) denotes <strong>the</strong> space <strong>of</strong> all order bounded operators.<br />

Clearly each regular operator is order bounded, that is, s c_ E b. The converse is not true <strong>in</strong><br />

general (see [29] for several counterexamples, <strong>the</strong> first <strong>of</strong> which was found by S. Kaplan).<br />

Here we can ask <strong>the</strong> question: when does <strong>the</strong> equality s = F b hold? The next famous<br />

<strong>the</strong>orem <strong>of</strong> E Riesz and L. Kantorovich describes a ra<strong>the</strong>r general situation when this<br />

happens.<br />

THEOREM 8 (Riesz-Kantorovich). Let X, Y be vector lattices with Y Dedek<strong>in</strong>d complete.<br />

Then E r (X, Y) = 12 b (X, Y), i.e., each order bounded operator T : X --+ Y is regular. Moreover,<br />

<strong>in</strong> this case, Er (X, Y) is itself a Dedek<strong>in</strong>d complete vector lattice whose lattice operations<br />

satisfy <strong>the</strong> Riesz-Kantorovich formulas, accord<strong>in</strong>g to which for each T ~ E r (X, Y)<br />

and each x ~ X+ we have<br />

T+(x) -- sup{Tu" 0

Positive operators 91<br />

(2) Is <strong>the</strong>re a non-Dedek<strong>in</strong>d complete vector lattice Y such that s (X, Y) is a vector<br />

lattice for each (some) vector lattice X? The answer is yes, and we refer to [18] for<br />

such examples and related discussions.<br />

(3) Assume that E r (X, Y) is a vector lattice, and so for each T E E r (X, Y) its modulus<br />

I TI exists. Is it true that this operator is necessarily given by <strong>the</strong> Riesz-Kantorovich<br />

formula? This is a long stand<strong>in</strong>g open problem.<br />

(4) Assume that for a fixed vector lattice Y <strong>the</strong> vector space s (X, Y) is a vector lattice<br />

for all vector lattices X <strong>in</strong> some special class <strong>of</strong> vector lattices. What impact does<br />

this have on <strong>the</strong> order structure <strong>of</strong> Y?<br />

In connection with (3) note that even when s (X, Y) is not assumed to be a vector<br />

lattice, <strong>the</strong>re is no s<strong>in</strong>gle example <strong>in</strong> <strong>the</strong> literature <strong>of</strong> a regular operator T:X --+ Y for<br />

which its modulus I TI exists and cannot be obta<strong>in</strong>ed by <strong>the</strong> Riesz-Kantorovich formula.<br />

Abramovich and Wickstead have studied this question ra<strong>the</strong>r systematically and described<br />

many cases covered by <strong>the</strong> Riesz-Kantorovich formula [ 18,19,140]. For <strong>the</strong> special doma<strong>in</strong><br />

<strong>of</strong> C(K)-vector lattices, this problem was also studied by van Rooij [118,119].<br />

Now let X and Y be <strong>Banach</strong> lattices. On <strong>the</strong> space E(X, Y) we will consider <strong>the</strong> usual<br />

operator norm. Under this norm <strong>the</strong> subspace s (X, Y) is rarely closed. However, <strong>the</strong>re<br />

is a natural norm on s (X, Y), <strong>the</strong> so-called regular norm, which is closely related to <strong>the</strong><br />

structure <strong>of</strong> <strong>the</strong> regular operators. The regular norm on s (X, Y) is def<strong>in</strong>ed by<br />

IlTllr - <strong>in</strong>f{ IISII" s E s<br />

Y) and S ~> iT}.<br />

Clearly II T II ~ IIT IIr and for positive operators both norms co<strong>in</strong>cide. It is well known that<br />

(Z; r (X, Y), II" IIr) is a <strong>Banach</strong> lattice. If Y is Dedek<strong>in</strong>d complete, <strong>the</strong>n <strong>the</strong> modulus Irl<br />

exists by Theorem 8, and <strong>in</strong> this case II r lit = IlITI II.<br />

3. When is every cont<strong>in</strong>uous operator regular?<br />

As <strong>the</strong> title <strong>of</strong> this section <strong>in</strong>dicates, we shall discuss here <strong>the</strong> pairs <strong>of</strong> <strong>Banach</strong> lattices X<br />

and Y for which <strong>the</strong> space <strong>of</strong> all cont<strong>in</strong>uous operators E(X, Y) co<strong>in</strong>cides with <strong>the</strong> space <strong>of</strong><br />

all regular operators s (X, Y). We should dist<strong>in</strong>guish here between two closely related but<br />

different questions. The first one requires simply <strong>the</strong> set-<strong>the</strong>oretic equality<br />

IZ(X, Y) = ~,r (x, Y), (*)<br />

while <strong>the</strong> second requires (,) and, additionally, <strong>the</strong> norm equality<br />

IIT II = II T Ilr for each T E L;(X, Y). (**)<br />

We will refer to <strong>the</strong>se problems as <strong>the</strong> isomorphic and isometric problems, respectively.<br />

The follow<strong>in</strong>g two <strong>the</strong>orems, which are essentially due to Kantorovich and Vulikh [76],<br />

describe <strong>the</strong> two most important classes <strong>of</strong> <strong>Banach</strong> lattices for which each cont<strong>in</strong>uous operator<br />

is regular. These are <strong>the</strong> familiar AM- and AL-spaces, also known under <strong>the</strong> names<br />

abstract M spaces and abstract L l spaces, respectively, and def<strong>in</strong>ed <strong>in</strong> [72, Section 5].

92 Y.A. Abramovich and C.D. Aliprantis<br />

THEOREM 9. Let Y be a Dedek<strong>in</strong>d complete <strong>Banach</strong> lattice with a strong unit. Then for<br />

each <strong>Banach</strong> lattice X equality (,) holds, that is, s Y) = s (X, Y). If <strong>the</strong> norm on Y<br />

is Fatou (<strong>in</strong> particular, if Y is equipped with a unit norm and, thus, is an AM-space), <strong>the</strong>n<br />

(**) also holds.<br />

THEOREM 10. If X is an AL-space and Y is an arbitrary <strong>Banach</strong> lattice with a Levi norm,<br />

<strong>the</strong>n equality (,) holds. If <strong>the</strong> norm on Y is Fatou, <strong>the</strong>n we also have (**).<br />

To be precise, it was assumed <strong>in</strong> [76] that Y <strong>in</strong> Theorem 10 was a KB-space, and it was<br />

noticed <strong>in</strong> [ 129] that <strong>the</strong> orig<strong>in</strong>al pro<strong>of</strong> could be easily carried over to an arbitrary <strong>Banach</strong><br />

lattice with a Levi norm. For a KB-space Y <strong>the</strong> pro<strong>of</strong>s can be found <strong>in</strong> [29, Theorem 15.3],<br />

and [ 134, Theorem 8.7.2]. Under <strong>the</strong> assumption that Y is positively complemented <strong>in</strong> Y**<br />

(which is somewhat stronger than <strong>the</strong> Levi property) a pro<strong>of</strong> <strong>of</strong> Theorem 10 is presented<br />

<strong>in</strong> [96, Theorem 1.5.11]. Two pr<strong>in</strong>cipal questions can be asked <strong>in</strong> connection with <strong>the</strong><br />

previous <strong>the</strong>orems.<br />

(1) Do Theorems 9 and 10 characterize <strong>the</strong> AL- and AM-spaces?<br />

(2) To what extend is <strong>the</strong> Levi condition essential for <strong>the</strong> validity <strong>of</strong> (,)?<br />

The first question is quite old and has various <strong>in</strong>terpretations depend<strong>in</strong>g on whe<strong>the</strong>r we<br />

deal with <strong>the</strong> isometric or <strong>the</strong> isomorphic version. We refer to [3] for a brief survey <strong>of</strong><br />

major results <strong>in</strong> this direction. Here we mention a few <strong>of</strong> <strong>the</strong>m, start<strong>in</strong>g with <strong>the</strong> isometric<br />

version <strong>of</strong> this question.<br />

THEOREM 1 1. Suppose that <strong>the</strong> <strong>Banach</strong> lattices X and Y satisfy both (,) and (**). Then<br />

ei<strong>the</strong>r X is an AL-space or Y is an AM-space.<br />

The isomorphic version is much deeper and has, <strong>in</strong> general, a negative solution [2].<br />

THEOREM 12. There exist two Dedek<strong>in</strong>d complete <strong>Banach</strong> lattices X and Y satisfy<strong>in</strong>g (,)<br />

such that X is not order isomorphic to any AL-space and Y is not order isomorphic to any<br />

AM-space. Moreover, for any given e > 0 <strong>the</strong> Dedek<strong>in</strong>d complete <strong>Banach</strong> lattices X and Y<br />

can be constructed <strong>in</strong> such a way that IIT IIr ~< (1 + e)II T ]l for each T ~ 12(X, Y).<br />

The strongest affirmative isomorphic result is also due to Cartwright and Lotz [44] and is<br />

stated next. A simple pro<strong>of</strong> <strong>of</strong> this result is given <strong>in</strong> [ 17], where <strong>the</strong> technique <strong>in</strong>troduced by<br />

Tzafriri <strong>in</strong> [133] <strong>of</strong> abstract Rademacher functions is utilized. The works <strong>of</strong> Freml<strong>in</strong> [59],<br />

~rno [ 105] and Schaefer [ 122] conta<strong>in</strong>ed some versions <strong>of</strong> <strong>the</strong> next result.<br />

THEOREM 13. Let two <strong>Banach</strong> lattices X and Y satisfy (,), and assume that ei<strong>the</strong>r X*<br />

or Y conta<strong>in</strong>s vector sublattices uniformly <strong>in</strong> n isomorphic to g n p for some p E [1 cx~). In<br />

<strong>the</strong> former case Y is order isomorphic to an AM-space and <strong>in</strong> <strong>the</strong> latter case X is order<br />

isomorphic to an AL-space.<br />

The case when X = Y is exceptionally nice and was studied <strong>in</strong> [2].<br />

THEOREM 14. If a <strong>Banach</strong> lattice X satisfies s<br />

ei<strong>the</strong>r to an AL- or AM-space.<br />

= s (X), <strong>the</strong>n X is order isomorphic

Positive operators 93<br />

PROOF. To sketch a pro<strong>of</strong>, note that <strong>the</strong> follow<strong>in</strong>g alternative is true: X ei<strong>the</strong>r conta<strong>in</strong>s<br />

uniformly <strong>in</strong> n <strong>the</strong> vector sublattices order isomorphic to g~ or it does not. In <strong>the</strong> former<br />

case, X* conta<strong>in</strong>s uniformly <strong>in</strong> n <strong>the</strong> vector sublattices g~ and so, by Theorem 13, X is<br />

isomorphic to an AM-space. In <strong>the</strong> latter case, X is isomorphic to an AL-space by Drno's<br />

<strong>the</strong>orem. []<br />

Questions (1) and (2) formulated above, though quite natural, have been addressed by<br />

Abramovich and Wickstead only recently <strong>in</strong> [21 ].<br />

THEOREM 15. A Dedek<strong>in</strong>d complete <strong>Banach</strong> lattice Y has a Levi norm if and only if for<br />

every AL-space X we have s Y) = s (X, Y).<br />

This converse to Theorem 10 is somewhat partial s<strong>in</strong>ce we assume Y to be Dedek<strong>in</strong>d<br />

complete. On <strong>the</strong> o<strong>the</strong>r hand, as <strong>the</strong> next two <strong>the</strong>orems <strong>in</strong> [21 ] show, it is <strong>the</strong> best one can<br />

get. Note also that <strong>the</strong>re are non-Dedek<strong>in</strong>d complete <strong>Banach</strong> lattices (hence, without a Levi<br />

norm) which, never<strong>the</strong>less, satisfy (,).<br />

THEOREM 16. For a <strong>Banach</strong> lattice Y <strong>the</strong> follow<strong>in</strong>g statements are equivalent:<br />

(a) Y is Dedek<strong>in</strong>d complete.<br />

(b) For all <strong>Banach</strong> lattices X, <strong>the</strong> space f_r (X, Y) is a Dedek<strong>in</strong>d complete vector lattice.<br />

(c) For all AL-spaces X, <strong>the</strong> space s (X, Y) is a vector lattice.<br />

THEOREM 17. For a <strong>Banach</strong> lattice Y <strong>the</strong> follow<strong>in</strong>g statements are equivalent:<br />

(1) Y is ~-Dedek<strong>in</strong>d complete.<br />

(2) s (X, Y) is ~-Dedek<strong>in</strong>d complete for each separable <strong>Banach</strong> lattice X.<br />

(3) s (c, Y) is a vector lattice, where c is <strong>the</strong> space <strong>of</strong> all convergent sequences.<br />

(4) /:r (LI [0, 27r], Y) is a vector lattice.<br />

We close this section by mention<strong>in</strong>g one more direction <strong>of</strong> research devoted to <strong>the</strong> connections<br />

between <strong>the</strong> order and topological properties <strong>of</strong> compact operators. Namely, what<br />

can be said about <strong>the</strong> modulus <strong>of</strong> a compact operator T :X --+ Y between <strong>Banach</strong> lattices?<br />

Does this modulus exist? When is it compact or weakly compact? A well known example<br />

by Krengel [29, Example 16.6] shows that even on ~2 <strong>the</strong> modulus <strong>of</strong> a compact operator<br />

may fail to exist. We refer to [20] for some history, fur<strong>the</strong>r references and concrete results<br />

concern<strong>in</strong>g <strong>the</strong> properties <strong>of</strong> <strong>the</strong> modulus <strong>of</strong> a compact operator.<br />

4. Dom<strong>in</strong>ation, compactness, and factorization<br />

DEFINITION 18. Let T, B : X --+ Y be two operators between vector lattices with B positive.<br />

We say that <strong>the</strong> operator T is dom<strong>in</strong>ated by <strong>the</strong> operator B (or that B dom<strong>in</strong>ates T)<br />

provided ]T(x)l ~< B(Ixl) for each x 6 X.<br />

It should be clear that every operator between <strong>Banach</strong> lattices dom<strong>in</strong>ated by a positive<br />

operator is automatically cont<strong>in</strong>uous, and that a positive operator T is dom<strong>in</strong>ated by ano<strong>the</strong>r<br />

positive operator B if and only if 0 ~< T ~< B. When Y is Dedek<strong>in</strong>d complete, an

94 Y.A. Abramovich and C.D. Aliprantis<br />

operator T is dom<strong>in</strong>ated by a positive operator B if and only if T is regular and I TI ~< B<br />

holds (use Theorem 8 to check this).<br />

We are now ready to discuss a new and <strong>in</strong>terest<strong>in</strong>g direction deal<strong>in</strong>g with dom<strong>in</strong>ated<br />

operators on <strong>Banach</strong> lattices. This direction can be described collectively as <strong>the</strong> dom<strong>in</strong>ation<br />

problem.<br />

THE DOMINATION PROBLEM. Assume that B : X --+ Y is a positive operator between <strong>Banach</strong><br />

lattices and assume that B satisfies some property ( P ). What effect does this property<br />

have on an operator T dom<strong>in</strong>ated by B ?<br />

Of course, <strong>the</strong> best one can expect is that T also satisfies (P). Here are a few examples<br />

<strong>of</strong> <strong>the</strong> properties that have been studied <strong>in</strong> connection with <strong>the</strong> dom<strong>in</strong>ation problem:<br />

compactness, weak compactness, Dunford-Pettis, and Radon-Nikodym.<br />

A first general dom<strong>in</strong>ation result was established <strong>in</strong> 1972 by Abramovich [ 1 ] for weakly<br />

compact operators. He proved that if a l<strong>in</strong>ear operator S : X --+ Y from a <strong>Banach</strong> lattice to<br />

a KB-space is dom<strong>in</strong>ated by a weakly compact operator, <strong>the</strong>n S is also weakly compact.<br />

Several years later Pitt [ 109] proved a dom<strong>in</strong>ation result for compact operators, namely he<br />

showedthatif B isacompactpositive operatoron Lp(#) with 1 < p < oo andO

Positive operators 95<br />

The exponents three and two <strong>in</strong> <strong>the</strong> previous two <strong>the</strong>orems are <strong>the</strong> best possible. Some<br />

historical comments regard<strong>in</strong>g this <strong>the</strong>orem and its predecessors can be found <strong>in</strong> [29,<br />

p. 279]. Are <strong>the</strong>re any o<strong>the</strong>r cases (not covered by <strong>the</strong> Dodds-Freml<strong>in</strong> Theorem) where<br />

one can deduce <strong>the</strong> compactness <strong>of</strong> <strong>the</strong> dom<strong>in</strong>ated operator itself? It turns out that <strong>the</strong>re<br />

are. This was shown by Wickstead <strong>in</strong> [142]. Before stat<strong>in</strong>g this result, let us rem<strong>in</strong>d <strong>the</strong><br />

reader that by a well known <strong>the</strong>orem <strong>of</strong> Walsh [ 137] <strong>the</strong> atomic <strong>Banach</strong> lattices with order<br />

cont<strong>in</strong>uous norm are precisely those <strong>Banach</strong> lattices <strong>in</strong> which every order <strong>in</strong>terval is norm<br />

compact, and that <strong>in</strong> such spaces <strong>the</strong> solid hull <strong>of</strong> any norm compact set is aga<strong>in</strong> norm<br />

compact [138, Theorem 5].<br />

THEOREM 22. For a pair <strong>of</strong> <strong>Banach</strong> lattices X and Y <strong>the</strong> follow<strong>in</strong>g two statements are<br />

equivalent:<br />

(1) Every operator from X to Y which is dom<strong>in</strong>ated by a compact operator is itself<br />

compact.<br />

(2) One <strong>of</strong> <strong>the</strong> follow<strong>in</strong>g three (non-exclusive) alternatives holds:<br />

(a) Both X* and Y have order cont<strong>in</strong>uous norm.<br />

(b) Y is an atomic <strong>Banach</strong> lattice with order cont<strong>in</strong>uous norm.<br />

(c) X* is an atomic <strong>Banach</strong> lattice with order cont<strong>in</strong>uous norm.<br />

A complete solution <strong>of</strong> <strong>the</strong> dom<strong>in</strong>ation problem for weakly compact operators preceded<br />

<strong>the</strong> solution <strong>of</strong> <strong>the</strong> compact case and was found also by Wickstead [ 139].<br />

THEOREM 23. For a pair <strong>of</strong> <strong>Banach</strong> lattices X and Y <strong>the</strong> follow<strong>in</strong>g two statements are<br />

equivalent:<br />

(1) Each operator from X to Y which is dom<strong>in</strong>ated by a weakly compact operator is<br />

weakly compact.<br />

(2) Ei<strong>the</strong>r X* or Y has order cont<strong>in</strong>uous norm.<br />

The follow<strong>in</strong>g analogue <strong>of</strong> Theorem 21 for weakly compact operators (that is, when no<br />

conditions on <strong>Banach</strong> lattices is imposed) was established <strong>in</strong> [26].<br />

THEOREM 24. Suppose that <strong>in</strong> <strong>the</strong> scheme X M~ y M2 Z <strong>of</strong> cont<strong>in</strong>uous operators between<br />

(real or complex) <strong>Banach</strong> lattices each operator Mi is dom<strong>in</strong>ated by a weakly compact<br />

positive operator, <strong>the</strong>n M2 Ml is a weakly compact operator. In particular, if an operator<br />

T : X --+ X on a <strong>Banach</strong> lattice is dom<strong>in</strong>ated by a weakly compact operator, <strong>the</strong>n T 2<br />

is weakly compact.<br />

Recall that a cont<strong>in</strong>uous operator T:X --+ Y between <strong>Banach</strong> spaces is said to be a<br />

Dunford-Pettis operator if T carries weakly precompact subsets <strong>of</strong> X onto norm precompact<br />

subsets <strong>of</strong> Y. The dom<strong>in</strong>ation problem for Dunford-Pettis operators was orig<strong>in</strong>ally<br />

considered by Aliprantis and Burk<strong>in</strong>shaw [27]. Improv<strong>in</strong>g <strong>the</strong> ma<strong>in</strong> result <strong>in</strong> [27], Kalton<br />

and Saab [73] proved <strong>the</strong> follow<strong>in</strong>g <strong>the</strong>orem.<br />

THEOREM 25. Let B : X --+ Y be a Dunford-Pettis operator between <strong>Banach</strong> lattices and<br />

assume that Y has order cont<strong>in</strong>uous norm. Then any operator T dom<strong>in</strong>ated by B is also<br />

Dunford-Pettis.

96 Y.A. Abramovich and C.D. Aliprantis<br />

Wickstead [ 142] showed that <strong>the</strong> only o<strong>the</strong>r case when <strong>the</strong> Dunford-Pettis property is<br />

<strong>in</strong>herited by a dom<strong>in</strong>ated operator is when X has weakly sequentially cont<strong>in</strong>uous lattice<br />

operations. The atomic <strong>Banach</strong> lattices with order cont<strong>in</strong>uous norm and <strong>the</strong> AM-spaces<br />

have this property. For such X <strong>the</strong> <strong>Banach</strong> lattice Y may be arbitrary. Some <strong>in</strong>terest<strong>in</strong>g<br />

results on <strong>the</strong> dom<strong>in</strong>ation problem for nonl<strong>in</strong>ear operators can be found <strong>in</strong> [38].<br />

Now we turn our attention to <strong>the</strong> factorization problem. Recall that a cont<strong>in</strong>uous operator<br />

T : X --+ Y between <strong>Banach</strong> spaces factors through a <strong>Banach</strong> space Z if <strong>the</strong>re exist<br />

cont<strong>in</strong>uous operators U:X --+ Z and V:Z --+ Y such that T -- V U. Depend<strong>in</strong>g on <strong>the</strong><br />

properties <strong>of</strong> <strong>the</strong> space Z and <strong>of</strong> <strong>the</strong> operators U and V, a lot can be learned about <strong>the</strong><br />

properties <strong>of</strong> <strong>the</strong> operator T. If <strong>the</strong> spaces under consideration are <strong>Banach</strong> lattices, <strong>the</strong>n we<br />

can immediately look at positive or regular operators between <strong>the</strong>m and demand more <strong>of</strong><br />

<strong>the</strong> factorization space Z and <strong>of</strong> <strong>the</strong> factor operators U and V.<br />

A classical example to illustrate this topic is <strong>the</strong> follow<strong>in</strong>g. If a cont<strong>in</strong>uous operator<br />

T:X --+ Y between <strong>Banach</strong> spaces factors through a reflexive <strong>Banach</strong> space, <strong>the</strong>n T is<br />

weakly compact. Remarkably, as shown by Davis, Figiel, Johnson, and Petczyfiski [50],<br />

this is a characterization <strong>of</strong> weakly compact operators.<br />

THEOREM 26. An operator between two <strong>Banach</strong> spaces is weakly compact if and only if<br />

factors through a reflexive <strong>Banach</strong> space.<br />

As a matter <strong>of</strong> fact, Theorem 26 can be considered as <strong>the</strong> start<strong>in</strong>g po<strong>in</strong>t for <strong>the</strong> subject<br />

<strong>of</strong> factorization. Without try<strong>in</strong>g to be complete on this topic, we will present here only a<br />

few results on factorization <strong>of</strong> operators between <strong>Banach</strong> lattices. Additional material on<br />

factorization <strong>of</strong> positive operators may be found <strong>in</strong> [28] and [96].<br />

Figiel [57] and Johnson [71 ] found which extra conditions on <strong>the</strong> factors and <strong>the</strong> transient<br />

space can be deduced for compact operators.<br />

THEOREM 27. Every compact operator between <strong>Banach</strong> spaces factors with compact factors<br />

through a separable reflexive <strong>Banach</strong> space.<br />

Inspired by <strong>the</strong>se results, Aliprantis and Burk<strong>in</strong>shaw [28] studied <strong>the</strong> factorization problem<br />

under <strong>the</strong> presence <strong>of</strong> order structures. The follow<strong>in</strong>g question is central for this topic.<br />

THE FACTORIZATION PROBLEM. Does every positive compact operator act<strong>in</strong>g between<br />

<strong>Banach</strong> lattices factor (with positive compact factors if possible) through a reflexive <strong>Banach</strong><br />

lattice?<br />

This problem (reiterated <strong>in</strong> [68] as Problem 9) is still open. We will mention some recent<br />

affirmative results below. But first, we should po<strong>in</strong>t out that a slightly weaker problem,<br />

when <strong>the</strong> given operator is assumed to be only weakly compact, is very difficult, too. It has<br />

been solved <strong>in</strong> <strong>the</strong> negative not a long time ago by Talagrand [130].<br />

EXAMPLE 28. There exists a positive weakly compact operator T:s<br />

cannot be factored through a reflexive <strong>Banach</strong> lattice.<br />

--~ C[0, 1] which

Positive operators 97<br />

The role <strong>of</strong> <strong>the</strong> space C[0, 1], or more generally, <strong>of</strong> <strong>the</strong> spaces C(K), for <strong>the</strong> factorization<br />

problem was studied <strong>in</strong> great detail by Ghoussoub [61 ] and Ghoussoub and Johnson <strong>in</strong> [62,<br />

63]. On a positive note regard<strong>in</strong>g <strong>the</strong> factorization problem, we will mention <strong>the</strong> follow<strong>in</strong>g<br />

<strong>the</strong>orem obta<strong>in</strong>ed <strong>in</strong> [28].<br />

THEOREM 29. The square <strong>of</strong> a positive compact (respectively weakly compact) operator<br />

on a <strong>Banach</strong> lattice factors through a reflexive <strong>Banach</strong> lattice with positive compact<br />

(respectively weakly compact) factors.<br />

From this <strong>the</strong>orem and Example 28 it follows that not each weakly compact operator<br />

is <strong>the</strong> square <strong>of</strong> a weakly compact operator. This, <strong>of</strong> course, follows from o<strong>the</strong>r known<br />

results as well. (For <strong>in</strong>stance, <strong>the</strong>re are cases when <strong>the</strong> square <strong>of</strong> a weakly compact operator<br />

is necessarily compact.) The next result by Aliprantis and Burk<strong>in</strong>shaw is a special<br />

case <strong>of</strong> Theorem 17.13 <strong>in</strong> [29]; it demonstrates a nice relation between factorization and<br />

dom<strong>in</strong>ation.<br />

THEOREM 30. Assume that T : X --+ X is a positive compact operator on a <strong>Banach</strong> lattice<br />

whose dual X* has order cont<strong>in</strong>uous norm. Then <strong>the</strong>re exist a reflexive <strong>Banach</strong> lattice Z<br />

and positive operators U : X --+ Z, V : Z --+ X such that T = V U. Moreover, any operator<br />

S : X --+ X which is dom<strong>in</strong>ated by <strong>the</strong> operator T also factors through Z and S -- V1U for<br />

some O

98 YA. Abramovich and C.D. Aliprantis<br />

An example <strong>in</strong> [110] shows that one cannot improve (3) by gett<strong>in</strong>g, for a positive regularly<br />

approximable T, both factors positive. A detailed study <strong>of</strong> when, <strong>in</strong> that example,<br />

both factors can be chosen positive is presented <strong>in</strong> [ 111 ].<br />

5. Invariant subspaces <strong>of</strong> positive operators<br />

In this section we will discuss <strong>the</strong> Invariant Subspace Problem for operators that are ei<strong>the</strong>r<br />

positive or closely associated with positive operators. The general <strong>the</strong>ory concern<strong>in</strong>g <strong>the</strong><br />

<strong>in</strong>variant subspace problem will be presented <strong>in</strong> a separate article prepared for this volume<br />

by Enflo and Lomonosov.<br />

THE INVARIANT SUBSPACE PROBLEM. Does a cont<strong>in</strong>uous l<strong>in</strong>ear operator T : X --+ X on<br />

a <strong>Banach</strong> space have a non-trivial closed <strong>in</strong>variant subspace?<br />

A vector subspace is "non-trivial" if it is different from {0} and X. A subspace V <strong>of</strong> X is<br />

T-<strong>in</strong>variant if T (V) _c V. If V is <strong>in</strong>variant under every cont<strong>in</strong>uous operator that commutes<br />

with T, <strong>the</strong>n V is called T-hyper<strong>in</strong>variant.<br />

If X is a f<strong>in</strong>ite dimensional complex <strong>Banach</strong> space <strong>of</strong> dimension greater than one,<br />

<strong>the</strong>n each non-zero operator T has a non-trivial closed <strong>in</strong>variant subspace. On <strong>the</strong> o<strong>the</strong>r<br />

hand, if X is non-separable, <strong>the</strong>n <strong>the</strong> closed vector subspace generated by <strong>the</strong> orbit<br />

{x, Tx, TZx .... } <strong>of</strong> any non-zero vector x is a non-trivial closed T-<strong>in</strong>variant subspace.<br />

Thus, <strong>the</strong> "<strong>in</strong>variant subspace problem" is <strong>of</strong> substance only when X is an <strong>in</strong>f<strong>in</strong>ite dimensional<br />

separable <strong>Banach</strong> space. Accord<strong>in</strong>gly, without any fur<strong>the</strong>r mention<strong>in</strong>g, all <strong>Banach</strong><br />

spaces under consideration <strong>in</strong> this section will be assumed to be <strong>in</strong>f<strong>in</strong>ite dimensional separable<br />

real or complex <strong>Banach</strong> spaces. The only exception will be made while discuss<strong>in</strong>g<br />

<strong>the</strong> Perron-Frobenius Theorem.<br />

In 1976, Enflo [56] was <strong>the</strong> first to construct an example <strong>of</strong> a cont<strong>in</strong>uous operator on<br />

a separable <strong>Banach</strong> space without a non-trivial closed <strong>in</strong>variant subspace, and thus he<br />

demonstrated that <strong>in</strong> this general form <strong>the</strong> <strong>in</strong>variant subspace problem has a negative answer.<br />

Subsequently, Read [ 115-117] has constructed a class <strong>of</strong> bounded operators on el<br />

without <strong>in</strong>variant subspaces. For operators on a Hilbert space, <strong>the</strong> existence <strong>of</strong> an <strong>in</strong>variant<br />

subspace is still unknown and is one <strong>of</strong> <strong>the</strong> famous unsolved problems <strong>of</strong> modern ma<strong>the</strong>matics.<br />

Due to <strong>the</strong> above counterexamples, <strong>the</strong> present study <strong>of</strong> <strong>the</strong> <strong>in</strong>variant subspace<br />

problem for operators on <strong>Banach</strong> spaces has been focused on various classes <strong>of</strong> operators<br />

for which one can expect <strong>the</strong> existence <strong>of</strong> an <strong>in</strong>variant subspace.<br />

We start our <strong>in</strong>variant subspace results with a version <strong>of</strong> <strong>the</strong> classical Perron-Frobenius<br />

<strong>the</strong>orem for positive matrices. As usual, we denote by r(T) <strong>the</strong> spectral radius <strong>of</strong> an operator<br />

T.<br />

THEOREM 3 2 (Perron-Frobenius). If A is a non-negative n x n matrix such that for some<br />

k >~ 1 <strong>the</strong> matrix A k has strictly positive entries, <strong>the</strong>n <strong>the</strong> spectral radius <strong>of</strong> A is a strictly<br />

positive eigenvalue <strong>of</strong> multiplicity one hav<strong>in</strong>g a strictly positive eigenvector.<br />

The pro<strong>of</strong> <strong>of</strong> this <strong>the</strong>orem, discovered by Frobenius [60] and Perron [108], is available<br />

<strong>in</strong> practically every book treat<strong>in</strong>g non-negative matrices, for <strong>in</strong>stance <strong>in</strong> [33,35,98]. One

Positive operators 99<br />

more pro<strong>of</strong> <strong>of</strong> <strong>the</strong> Perron-Frobenius <strong>the</strong>orem as well as many <strong>in</strong>terest<strong>in</strong>g generalizations<br />

1<br />

can be found <strong>in</strong> [ 112]. If all entries <strong>of</strong> A are strictly positive, <strong>the</strong>n <strong>the</strong> sequence { [r(a)] k Aku}<br />

converges to a unique strictly positive eigenvector correspond<strong>in</strong>g to <strong>the</strong> eigenvalue r (A), no<br />

matter which <strong>in</strong>itial vector u > 0 is chosen. This fact has numerous applications. A major<br />

step <strong>in</strong> extend<strong>in</strong>g <strong>the</strong> Perron-Frobenius Theorem to <strong>in</strong>f<strong>in</strong>ite dimensional sett<strong>in</strong>gs was done<br />

by Kre<strong>in</strong> and Rutman [82] who proved <strong>the</strong> follow<strong>in</strong>g <strong>the</strong>orem.<br />

THEOREM 3 3 (Kre<strong>in</strong>-Rutman). For any positive operator T" X --~ X on a <strong>Banach</strong> lattice<br />

r(T) E or(T), i.e., <strong>the</strong> spectral radius <strong>of</strong> T belongs to <strong>the</strong> spectrum <strong>of</strong> T. Fur<strong>the</strong>rmore, if<br />

T is also compact and r(T) > O, <strong>the</strong>n <strong>the</strong>re exists some x > 0 such that Tx -- r(T)x.<br />

PROOF. We will sketch a pro<strong>of</strong>. The <strong>in</strong>clusion r(T) E cr(T) is caused merely by <strong>the</strong> positivity<br />

<strong>of</strong> T. Indeed, if we denote by R()~) <strong>the</strong> resolvent operator <strong>of</strong> T, <strong>the</strong>n clearly R0~) > 0<br />

for each )~ > r(T), Also for each )~ with 1)~1 > r(T) <strong>the</strong> <strong>in</strong>equality ]]R(])~[)]] ~> ]]R()~)]]<br />

1<br />

holds. Therefore, for )~, -- r(T) + -d we have []R()~n)l[ ~ oc, whence r(T) ~ or(T).<br />

Assume fur<strong>the</strong>r that T is compact and r(T) > 0. There exist unit vectors Yn E X+<br />

such that ]lRO~,)yn I[--~ oc. Us<strong>in</strong>g <strong>the</strong> vectors y,, we <strong>in</strong>troduce <strong>the</strong> unit vectors xn =<br />

R()~n)yn/llR()~n)Yn][ ~ X+. S<strong>in</strong>ce T is compact we can assume that Txn --+ x ~ X+. F<strong>in</strong>ally,<br />

us<strong>in</strong>g <strong>the</strong> identities<br />

r(T)xn - Txn -<br />

[r(T)- Xn]xn + ()~n - T)xn<br />

= [r(T) - ~,]x, + y~/IIRO~)y~II<br />

and that r(T) > 0 we <strong>in</strong>fer that x 5~ 0 and that Tx -- r(T)x.<br />

The conclusion <strong>of</strong> <strong>the</strong> previous <strong>the</strong>orem rema<strong>in</strong>s valid if we replace <strong>the</strong> compactness<br />

<strong>of</strong> T by <strong>the</strong> compactness <strong>of</strong> some power <strong>of</strong> T. Indeed, assume that T k is compact for<br />

some k. S<strong>in</strong>ce r(T k) = [r(T)] k > 0 <strong>the</strong> previous <strong>the</strong>orem implies that <strong>the</strong>re is a vector<br />

x > 0 such that Tkx = [r(T)]kx. It rema<strong>in</strong>s to verify that <strong>the</strong> non-zero positive vector<br />

Y -- Y~i=0 k-I r iTk-l-i x is an eigenvector <strong>of</strong> T correspond<strong>in</strong>g to <strong>the</strong> eigenvalue r(T).<br />

The reader is referred to [121,123,144] for complete pro<strong>of</strong>s and many pert<strong>in</strong>ent results<br />

concern<strong>in</strong>g <strong>the</strong> Kre<strong>in</strong>-Rutman <strong>the</strong>orem. Some relevant results can be found <strong>in</strong> [4]. Note<br />

that <strong>the</strong> Kre<strong>in</strong>-Rutman <strong>the</strong>orem holds not only for <strong>Banach</strong> lattices but for ordered <strong>Banach</strong><br />

spaces as well. There is an <strong>in</strong>terest<strong>in</strong>g approach allow<strong>in</strong>g to relax <strong>the</strong> compactness assumption.<br />

Namely, as shown by Zabrel3cO and Smickih [146] and <strong>in</strong>dependently by Nussbaum<br />

[ 102], <strong>in</strong>stead <strong>of</strong> <strong>the</strong> compactness <strong>of</strong> T it is enough to assume only that <strong>the</strong> essential<br />

spectral radius re(T) is strictly less than <strong>the</strong> spectral radius r(T). A different type <strong>of</strong> relaxation<br />

is considered <strong>in</strong> [121], where <strong>the</strong> restriction <strong>of</strong> T to X+ is assumed to be compact,<br />

that is, T maps <strong>the</strong> positive part <strong>of</strong> <strong>the</strong> unit ball <strong>in</strong>to a precompact set. A version <strong>of</strong> this<br />

result, given <strong>in</strong> terms <strong>of</strong> re(T), can be found <strong>in</strong> [ 102].<br />

Ano<strong>the</strong>r classical result by M. Kre<strong>in</strong> [82, Theorem 6.3] is <strong>the</strong> follow<strong>in</strong>g.<br />

THEOREM 34 (Kre<strong>in</strong>). Let T" C(I-2) --+ C(I-2) be a positive operator, where S72 is a compact<br />

Hausdorff space. Then T*, <strong>the</strong> adjo<strong>in</strong>t <strong>of</strong> T, has a positive eigenvector correspond<strong>in</strong>g<br />

to a non-negative eigenvalue.

100 Y.A. Abramovich and C.D. Aliprantis<br />

PROOF. Consider <strong>the</strong> set G -- {f E C(t'2)+ f(1) = 1}, where 1 denotes <strong>the</strong> constant<br />

function one on t'2. Clearly, G is a nonempty, convex, and w*-compact subset <strong>of</strong> C(~)*.<br />

Next, def<strong>in</strong>e <strong>the</strong> mapp<strong>in</strong>g F'G --+ G by<br />

f+T*f f+T*f<br />

F(f) -- =<br />

[f + T'f](1) 1 + f(rl)<br />

A straightforward verification shows that F <strong>in</strong>deed maps <strong>the</strong> set G <strong>in</strong>to itself and that<br />

F: (G, w*) --+ (G, w*) is a cont<strong>in</strong>uous function. So, by Tychon<strong>of</strong>f's fixed po<strong>in</strong>t <strong>the</strong>orem<br />

(see, for <strong>in</strong>stance [22, Corollary 16.52]) <strong>the</strong>re exists some q~ E G such that F(4~) = 4~. That<br />

is, q~ + T*4~ -- [1 + 4~(T1)]4), or T*4) -- 4)(T1)4~, establish<strong>in</strong>g that 0 < 4~ E C(12)~_ is an<br />

eigenvector for T* hav<strong>in</strong>g <strong>the</strong> non-negative eigenvalue 4~(T1).<br />

M<br />

A pro<strong>of</strong> <strong>of</strong> Theorem 34 that does not use fixed po<strong>in</strong>t <strong>the</strong>orems can be found <strong>in</strong> [77].<br />

COROLLARY 3 5. Every positive operator on a C (Y2)-space (where Y2 is Hausdorff, compact<br />

and not a s<strong>in</strong>gleton) which is not a multiple <strong>of</strong> <strong>the</strong> identity has a non-trivial hyper<strong>in</strong>variant<br />

closed subspace.<br />

PROOF. Let T'C(S2) --+ C(Y2) be a positive operator which is not a multiple <strong>of</strong> <strong>the</strong> identity.<br />

By Theorem 34 <strong>the</strong> adjo<strong>in</strong>t operator T* has a positive eigenvector. If )~ denotes <strong>the</strong><br />

correspond<strong>in</strong>g eigenvalue, <strong>the</strong>n <strong>the</strong> subspace (T - )~I)(X) has <strong>the</strong> desired properties. V]<br />

Recall that a cont<strong>in</strong>uous operator T : X ~ X on a <strong>Banach</strong> space is said to be quas<strong>in</strong>ilpotent<br />

if its spectral radius is zero. It is well known that T is quas<strong>in</strong>ilpotent if and only if<br />

limn~ ]lTnxl] 1/n = 0 for eachx c X. It can happen thata cont<strong>in</strong>uous operator T: X ~ X<br />

is not quas<strong>in</strong>ilpotent but, never<strong>the</strong>less, limn~ IlTnx ]ll/n = 0 for some x ~ 0. In this case<br />

we say that T is locally quas<strong>in</strong>ilpotent at x. This property was <strong>in</strong>troduced <strong>in</strong> [6], where it<br />

was found to be useful <strong>in</strong> <strong>the</strong> study <strong>of</strong> <strong>the</strong> <strong>in</strong>variant subspace problem. The set <strong>of</strong> po<strong>in</strong>ts at<br />

which T is quas<strong>in</strong>ilpotent is denoted by QT, i.e., QT = {x 6 X: limn~ IlTnxll 1/n =0}.<br />

It is easy to prove that <strong>the</strong> set QT is a T-hyper<strong>in</strong>variant vector subspace. We formulate<br />

below a few simple properties <strong>of</strong> <strong>the</strong> vector space QT.<br />

9 The operator T is quas<strong>in</strong>ilpotent if and only if QT = X.<br />

9 QT = {0} is possible every isometry T satisfies QT = {0}. Notice also that even<br />

a compact positive operator can fail to be locally quas<strong>in</strong>ilpotent at every non-zero<br />

vector9 For <strong>in</strong>stance, consider <strong>the</strong> compact positive operator T:g2 -+ ~2 def<strong>in</strong>ed by<br />

T(xl, X2, 9 9 .) -- (X l, x2 2 ' x3 3 .... ). For each non-zero x E ~2 pick some k for which Xk r 0<br />

and note that IlTnxl] 1/n ~ ~]xk[ 1/n for each n, from which it follows that T is not<br />

quas<strong>in</strong>ilpotent at x.<br />

9 QT can be dense without be<strong>in</strong>g equal to X. For <strong>in</strong>stance, <strong>the</strong> left shift S: g2 -+ g2, def<strong>in</strong>ed<br />

by S(xl, x2, x3 .... ) = (x2, x3 .... ), has this property.<br />

9 If QT ~: {0} and QT ~ X, <strong>the</strong>n QT is a non-trivial closed T-hyper<strong>in</strong>variant subspace<br />

<strong>of</strong> X.<br />

The above properties show that as far as <strong>the</strong> <strong>in</strong>variant subspace problem is concerned,<br />

we need only consider <strong>the</strong> two extreme cases: QT = {0} and QT = X.

Positive operators 101<br />

We are now ready to state <strong>the</strong> ma<strong>in</strong> result about <strong>the</strong> existence <strong>of</strong> <strong>in</strong>variant subspaces<br />

<strong>of</strong> positive operators on gp-spaces. It implies, <strong>in</strong> particular, that if a positive operator is<br />

quas<strong>in</strong>ilpotent at a non-zero positive vector, <strong>the</strong>n <strong>the</strong> operator has an <strong>in</strong>variant subspace.<br />

This is an improvement <strong>of</strong> <strong>the</strong> ma<strong>in</strong> result <strong>in</strong> [6].<br />

THEOREM 36. Let T : g~p --+ g~p (1

102 YA. Abramovich and C.D. Aliprantis<br />

A very <strong>in</strong>terest<strong>in</strong>g open problem related to Corollary 39 is whe<strong>the</strong>r or not each positive<br />

operator on el has a nontrivial closed <strong>in</strong>variant subspace. S<strong>in</strong>ce each cont<strong>in</strong>uous operator<br />

on el has a modulus (see Theorem 10), a natural candidate to test this problem is <strong>the</strong><br />

modulus <strong>of</strong> any operator on g l without a nontrivial closed <strong>in</strong>variant subspace. In particular,<br />

each operator on g 1 without <strong>in</strong>variant subspace constructed by Read [ 115,117] is such a<br />

candidate. Troitsky [ 131 ] has recently managed to handle <strong>the</strong> case <strong>of</strong> <strong>the</strong> quas<strong>in</strong>ilpotent operator<br />

T constructed by Read <strong>in</strong> [ 117]. Not only does I TI have a nontrivial closed <strong>in</strong>variant<br />

subspace, but I TI also has a positive eigenvector.<br />

In our previous discussion, we were consider<strong>in</strong>g only operators on gp-spaces. However,<br />

we only used <strong>the</strong> discreteness <strong>of</strong> g p-spaces, <strong>the</strong> above results rema<strong>in</strong> true for operators<br />

on arbitrary discrete <strong>Banach</strong> lattices, 3 <strong>in</strong> particular, for operators on Lorentz and Orlicz<br />

sequence spaces. For <strong>in</strong>stance, <strong>the</strong> follow<strong>in</strong>g analogue <strong>of</strong> Theorem 36 is true.<br />

THEOREM 40. Let T : X --+ X be a cont<strong>in</strong>uous operator with modulus, where X is a discrete<br />

<strong>Banach</strong> lattice. If <strong>the</strong>re exists a non-zero positive operator S : X -+ X which is locally<br />

quas<strong>in</strong>ilpotent at a non-zero positive vector and SITI

Positive operators 103<br />

<strong>the</strong> next two results. The pro<strong>of</strong> <strong>of</strong> <strong>the</strong> first one was <strong>in</strong>spired by Hilden's pro<strong>of</strong> (<strong>in</strong>cluded<br />

<strong>in</strong> [97]) <strong>of</strong> a simplified version <strong>of</strong> Lomonosov's <strong>in</strong>variant subspace <strong>the</strong>orem mentioned<br />

above.<br />

THEOREM 42. Let B : X --+ X be a positive operator on a <strong>Banach</strong> lattice. Assume that<br />

<strong>the</strong>re exists a positive operator S : X --+ X such that S B

104 Y.A. Abramovich and C.D. Aliprantis<br />

where K0 is compact. Then <strong>the</strong> required "triplet" (R, C, K) is <strong>the</strong> follow<strong>in</strong>g: R = B, C =<br />

B and K = K0.<br />

The pro<strong>of</strong> that each positive <strong>in</strong>tegral operator is compact-friendly will be given <strong>in</strong> Theorem<br />

52, and <strong>the</strong> pro<strong>of</strong> for composition operators can be found <strong>in</strong> [ 11 ].<br />

We will also need <strong>the</strong> follow<strong>in</strong>g ref<strong>in</strong>ement <strong>of</strong> <strong>the</strong> concept <strong>of</strong> <strong>the</strong> strong unit <strong>in</strong>troduced<br />

at <strong>the</strong> end <strong>of</strong> Section 1. A positive element u <strong>in</strong> a <strong>Banach</strong> lattice X is called a quasi-<strong>in</strong>terior<br />

po<strong>in</strong>t whenever Xu is norm dense <strong>in</strong> X, i.e., Xu -- X.<br />

LEMMA 45. If u is quasi-<strong>in</strong>terior po<strong>in</strong>t <strong>in</strong> a <strong>Banach</strong> lattice X, <strong>the</strong>n<br />

(1) For every non-zero element y ~ Xu <strong>the</strong>re exists an operator V : X ~ X dom<strong>in</strong>ated<br />

by <strong>the</strong> identity operator and such that Vy > O.<br />

(2) For every element v satisfy<strong>in</strong>g 0 0. Put<br />

M1 -- VI C, and note that M1 is dom<strong>in</strong>ated both by <strong>the</strong> compact positive operator K and<br />

by <strong>the</strong> operator R.<br />

From Jx2 = X and C ~-0, we see that <strong>the</strong>re exists some 0 < y

Positive operators 105<br />

ICyl 0 and which is dom<strong>in</strong>ated by both<br />

<strong>the</strong> positive compact operator K A and by RA.<br />

From M3 M2 M1 x 1 : M3x3 > 0, we see that M3 M2 M1 is a non-zero operator which (by<br />

Theorem 20) is also compact. Moreover, an easy argument shows that for each x 6 X<br />

IM3M2M1 (x)[ ~ RARAR(Ixl) ~ [RARAR + T](Ixl).<br />

Now consider <strong>the</strong> non-zero positive operator S -- RARAR + T. Then B and S commute,<br />

S dom<strong>in</strong>ates <strong>the</strong> non-zero compact operator M3MzM1, and B is quas<strong>in</strong>ilpotent at<br />

x0. By Theorem 43, S and B have a common non-trivial closed <strong>in</strong>variant ideal. This ideal<br />

is <strong>in</strong>variant under both B and T. [2<br />

We will illustrate this <strong>the</strong>orem while study<strong>in</strong>g <strong>in</strong>tegral operators <strong>in</strong> <strong>the</strong> next section. Theorem<br />

46 may suggest that <strong>the</strong> compact-friendly operators characterize <strong>the</strong> positive operators<br />

with an <strong>in</strong>variant subspace. However, this is not true. An example <strong>of</strong> a positive operator<br />

that is not compact-friendly but has a non-trivial <strong>in</strong>variant subspace is given <strong>in</strong> [ 11 ]. The<br />

operator itself is very simple, just a multiplication operator M~; but <strong>the</strong> fact that for some<br />

multipliers <strong>the</strong> operator is not compact-friendly is far from be<strong>in</strong>g trivial. As shown <strong>in</strong> [ 13],<br />

a multiplication operator M~o on an Lp(#) is compact-friendly if and only if <strong>the</strong> multiplier<br />

q) is constant on a set <strong>of</strong> positive measure. Never<strong>the</strong>less, <strong>in</strong> <strong>the</strong> last section <strong>of</strong> [ 11],<br />

a program is outl<strong>in</strong>ed for how <strong>the</strong> compact-friendly operators may be used to <strong>in</strong>clude <strong>the</strong><br />

exceptional cases as well.<br />

We conclude this section by mention<strong>in</strong>g <strong>the</strong> follow<strong>in</strong>g <strong>in</strong>terest<strong>in</strong>g conjecture <strong>of</strong> Lomonosov<br />

[89]: The adjo<strong>in</strong>t <strong>of</strong> an arbitrary operator on a <strong>Banach</strong> space has a non-trivial<br />

closed <strong>in</strong>variant subspace. For operators on <strong>Banach</strong> lattices this conjecture is also open.<br />

We refer to <strong>the</strong> papers by de Branges [36], Abramovich, Aliprantis and Burk<strong>in</strong>shaw [10]<br />

and Simoni6 [127,128] that deal with this and related problems.<br />

7. Integral operators and <strong>in</strong>variant subspaces<br />

After a brief <strong>in</strong>troduction to <strong>the</strong> <strong>the</strong>ory <strong>of</strong> <strong>in</strong>tegral operators, we will prove <strong>in</strong> Theorem 52<br />

that each positive <strong>in</strong>tegral operator is compact-friendly. After that, apply<strong>in</strong>g <strong>the</strong> <strong>the</strong>ory<br />

<strong>of</strong> compact-friendly operators, we will establish <strong>in</strong> Theorem 54 <strong>the</strong> existence <strong>of</strong> a nontrivial<br />

<strong>in</strong>variant subspace for each locally quas<strong>in</strong>ilpotent positive <strong>in</strong>tegral operator. The<br />

rest <strong>of</strong> <strong>the</strong> section will be devoted to a discussion <strong>of</strong> <strong>the</strong> And6-Krieger <strong>the</strong>orem and its<br />

generalizations.<br />

We beg<strong>in</strong> by def<strong>in</strong><strong>in</strong>g <strong>the</strong> classes <strong>of</strong> <strong>in</strong>tegral (= kernel) and almost <strong>in</strong>tegral operators. 5<br />

Throughout this section X and Y will denote two Dedek<strong>in</strong>d complete <strong>Banach</strong> lattices which<br />

5 To avoid a possible ambiguity let us po<strong>in</strong>t out that <strong>the</strong> <strong>in</strong>tegral operators considered <strong>in</strong> this section should not<br />

be confused with <strong>the</strong> <strong>in</strong>tegral operators <strong>in</strong> <strong>the</strong> sense <strong>of</strong> Gro<strong>the</strong>ndieck discussed <strong>in</strong> [72].

106 Y.A. Abramovich and C.D. Aliprantis<br />

are order dense ideals <strong>in</strong> <strong>the</strong> spaces (<strong>of</strong> equivalence classes) <strong>of</strong> all measurable functions<br />

L0(I2j, El, #1) and L0(f22,272, #2) respectively, where (I21,271, #1) and (Y22, I72, #2)<br />

are two arbitrary "non-pathological" measure spaces, for <strong>in</strong>stance, a-f<strong>in</strong>ite measure spaces.<br />

DEFINITION 47. An operator T" X --+ Y is said to be an <strong>in</strong>tegral operator if <strong>the</strong>re exists<br />

a #1 x #z-measurable function T (., .) on s x ~(22 such that for each x e X we have<br />

Tx(t) -- f~ T(s,t)x(s)dlzl(s)<br />

for #2-almost all t e S22.<br />

1<br />

The function T(., .) is usually referred to as <strong>the</strong> kernel <strong>of</strong> <strong>the</strong> operator T. It is easy<br />

to prove that an <strong>in</strong>tegral operator is positive if and only if its kernel is nonnegative almost<br />

everywhere on S21 x I22. Fur<strong>the</strong>rmore, it is well known that an <strong>in</strong>tegral operator T<br />

with kernel T (., .) is a regular operator from X to Y if and only if for each x e X <strong>the</strong><br />

function f IT(s, .)llx(s)l d#l (s) belongs to Y. Moreover, under this condition <strong>the</strong> modulus<br />

[TI:X --+ Y is also an <strong>in</strong>tegral operator, and its kernel is given by <strong>the</strong> absolute value<br />

IT (s, t)[ <strong>of</strong> <strong>the</strong> <strong>in</strong>itial kernel. It should be po<strong>in</strong>ted out that many authors <strong>in</strong>clude <strong>the</strong> last<br />

condition <strong>in</strong> <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> <strong>the</strong> <strong>in</strong>tegral operator, thus guarantee<strong>in</strong>g that <strong>the</strong> modulus I TI<br />

<strong>of</strong> <strong>the</strong> operator T also acts from X to Y. To stress this property, such operators are sometimes<br />

referred to as absolute <strong>in</strong>tegral operators.<br />

We refer to [41], [74, Chapter 11] and [144, Chapter 13] for a systematic study <strong>of</strong> <strong>in</strong>tegral<br />

operators with emphasis on <strong>the</strong> order structure <strong>of</strong> <strong>the</strong> spaces <strong>in</strong>volved. Apart from <strong>the</strong>se<br />

three books <strong>the</strong>re are, <strong>of</strong> course, many o<strong>the</strong>r books devoted to <strong>in</strong>tegral operators; see for<br />

<strong>in</strong>stance [51,54,67,75,78]. Whenever convenient we will identify an <strong>in</strong>tegral operator with<br />

its kernel.<br />

The <strong>in</strong>tegral operators are closely related to f<strong>in</strong>ite-rank operators. A rank-one operator<br />

from a <strong>Banach</strong> lattice E to a <strong>Banach</strong> lattice F is any operator <strong>of</strong> <strong>the</strong> form 4~ @ Y, where 4~ e<br />

E*, y e F and ~b | y (x) -- 4~ (x)y for each x e E. Any operator <strong>of</strong> <strong>the</strong> form Y~i=ln ~bi | Yi<br />

is called a f<strong>in</strong>ite-rank operator. We denote by E* | F <strong>the</strong> vector space <strong>of</strong> all f<strong>in</strong>ite rank<br />

operators from E to F. The f<strong>in</strong>ite-rank operators are precisely <strong>the</strong> cont<strong>in</strong>uous operators<br />

whose ranges are f<strong>in</strong>ite dimensional.<br />

Recall that <strong>the</strong> subspace <strong>of</strong> all order cont<strong>in</strong>uous l<strong>in</strong>ear functionals on E is denoted by E*.<br />

DEFINITION 48. A bounded operator T'E ~ F between <strong>Banach</strong> lattices, with F<br />

Dedek<strong>in</strong>d complete, is said to be an almost <strong>in</strong>tegral operator if T belongs to <strong>the</strong> band<br />

generated <strong>in</strong> Z2 r (E, F) by <strong>the</strong> collection E* | F <strong>of</strong> all f<strong>in</strong>ite-rank operators.<br />

The term "almost <strong>in</strong>tegral operator" has its orig<strong>in</strong> <strong>in</strong> Theorem 50 below that is due to<br />

Lozanovsky [90], who <strong>in</strong>troduced both this class <strong>of</strong> operators and <strong>the</strong> term itself. Accord<strong>in</strong>g<br />

to this <strong>the</strong>orem <strong>the</strong> classes <strong>of</strong> almost <strong>in</strong>tegral and <strong>in</strong>tegral operators are essentially <strong>the</strong> same.<br />

Let us mention also <strong>the</strong> paper by Synnatzschke [129], where a ref<strong>in</strong>ed version <strong>of</strong> almost<br />

<strong>in</strong>tegral operators is studied. Namely, one can fix arbitrary subsets N _ E* and M c_ F,<br />

and <strong>the</strong>n def<strong>in</strong>e <strong>the</strong> band <strong>of</strong> almost <strong>in</strong>tegral operators (which depends on N and M) as <strong>the</strong>

Positive operators 107<br />

band generated <strong>in</strong>/~r (E, F) by {q) | y" q9 E N, y E M}. The case <strong>of</strong> N = E* and M -- F<br />

considered <strong>in</strong> Def<strong>in</strong>ition 48 is <strong>the</strong> most important.<br />

A key element to Lozanovsky's characterization <strong>of</strong> almost <strong>in</strong>tegral operators is <strong>the</strong> famous<br />

Dunford <strong>the</strong>orem (see Theorem 49) regard<strong>in</strong>g <strong>the</strong> <strong>in</strong>tegral representation <strong>of</strong> operators<br />

from L1 <strong>in</strong>to Lp [55]. There are many pro<strong>of</strong>s <strong>of</strong> this important <strong>the</strong>orem. A simple pro<strong>of</strong>,<br />

utiliz<strong>in</strong>g <strong>the</strong> work <strong>of</strong> Benedek and Panzone [34] on spaces with mixed norm, was also<br />

found by Lozanovsky [91 ]. To outl<strong>in</strong>e <strong>the</strong> pro<strong>of</strong> we need some term<strong>in</strong>ology and notation.<br />

For p E (1, ec), let q denote its conjugate exponent, that is, p-1 + q-1 _ 1. We denote<br />

by L(p, oc) and L(q, 1) <strong>the</strong> usual <strong>Banach</strong> spaces with mixed norm [34], which consist <strong>of</strong><br />

all real-valued #l x #2-measurable functions f(s, t) on I21 x I22 satisfy<strong>in</strong>g respectively<br />

<strong>the</strong> follow<strong>in</strong>g conditions:<br />

and<br />

Ilfllp,oc "-Illlfllpll~- esssup<br />

s ES22 1<br />

If(s,t)lPd#2(t)<br />

]'J'<br />

108 Y.A. Abramovich and C.D. Aliprantis<br />

THEOREM 50 (Lozanovsky). An operator T : X--+ Y between two <strong>Banach</strong> function<br />

spaces is an almost <strong>in</strong>tegral operator if and only if T is a regular <strong>in</strong>tegral operator Fur<strong>the</strong>rmore,<br />

for this operator <strong>the</strong>re are measurable functions wi ~ Lo(#i) (i = 1, 2) and a<br />

unique (up to an equivalence) <strong>in</strong>tegrable function T(s, t) on 121 x ~C22 such that for each<br />

x ~ X and for lzz-almost all t ~ ~-22 <strong>the</strong> follow<strong>in</strong>g representation holds:<br />

Tx(t) = w2(t) fs2 T(s, t)x(s)wl (s) d/zl (s).<br />

1<br />

The idea <strong>of</strong> <strong>the</strong> pro<strong>of</strong> lies <strong>in</strong> consecutive approximations obta<strong>in</strong>ed by "squeez<strong>in</strong>g" X and<br />

Y between weighted L~ and L1 spaces (this procedure is described <strong>in</strong> [72, Section 5]) and<br />

<strong>the</strong>n apply<strong>in</strong>g <strong>the</strong> Dunford <strong>the</strong>orem.<br />

The previous <strong>the</strong>orem is a global order-<strong>the</strong>oretic characterization <strong>of</strong> <strong>the</strong> space <strong>of</strong> <strong>in</strong>tegral<br />

operators. An important <strong>in</strong>ternal characterization <strong>of</strong> an <strong>in</strong>dividual <strong>in</strong>tegral operator is due<br />

to A. Bukhvalov [37,41], see also [125], and it is given next. Many applications <strong>of</strong> this<br />

criterion can be found <strong>in</strong> [41 ].<br />

THEOREM 51. An operator T : X --+ Y between two <strong>Banach</strong> function spaces is an <strong>in</strong>tegral<br />

operator if and only if for every order bounded sequence {Xn } C X which converges to zero<br />

<strong>in</strong> measure, Txn ~ 0 holds #2-almost everywhere.<br />

THEOREM 5 2. Every positive <strong>in</strong>tegral operator T is compact-friendly.<br />

PROOF. As we already know T is an almost <strong>in</strong>tegral operator. Now note that for each<br />

almost <strong>in</strong>tegral operator <strong>the</strong>re exists a net { T~ } <strong>of</strong> positive operators such that 0 ~< T~ x t Tx<br />

holds for each x ) 0, and each Tc~ is dom<strong>in</strong>ated by a positive f<strong>in</strong>ite-rank operator. This<br />

implies that T is compact-friendly.<br />

D<br />

LEMMA 5 3. If S : X --+ X is a strictly positive <strong>in</strong>tegral operator on an order complete<br />

<strong>Banach</strong> lattice, <strong>the</strong>n for each element xo > 0 <strong>the</strong>re exists a compact positive operator<br />

K: X --+ X satisfy<strong>in</strong>g 0 ~ K ~ S 3 and K xo > O.<br />

PROOF. Let S:X ---> X be a strictly positive <strong>in</strong>tegral operator and fix x0 > 0. So, <strong>the</strong>re<br />

exists a net { S~ } <strong>of</strong> positive operators such that 0 0, we see that <strong>the</strong>re exists some <strong>in</strong>dex ~l such that S~x0 > 0<br />

for each c~/> ~l. Similarly, from S~(S~,xo) ~ S(S~,xo) and S(S~,xo) > 0, we see that<br />

S~(S~xo) > 0 for all ~ t> Ot2 ) Otl. F<strong>in</strong>ally, from <strong>the</strong> facts that S(S~2S~lXO) > 0 and<br />

S~(S~2S~lXO) t~ S(S~2S~,xo), it follows that S~(S~2S~,xo) > 0 for all possible <strong>in</strong>dices<br />

O~ /) O~ 3 /) Ot2 )/Otl. It rema<strong>in</strong>s to note that if we let K -- $33 , <strong>the</strong>n K" X ~ X is a positive<br />

compact operator satisfy<strong>in</strong>g 0 ~< K ~< S 3 and K xo > O.<br />

And now we are ready to present one <strong>of</strong> <strong>the</strong> ma<strong>in</strong> <strong>in</strong>variant subspace <strong>the</strong>orems for <strong>in</strong>tegral<br />

operators obta<strong>in</strong>ed <strong>in</strong> [7]. The pro<strong>of</strong> illustrates <strong>the</strong> role <strong>of</strong> compact-friendly operators.<br />

D

Positive operators 109<br />

THEOREM 54. Let S : X --+ X be a non-zero positive <strong>in</strong>tegral operator on a Dedek<strong>in</strong>d<br />

complete <strong>Banach</strong> lattice and let B : X --+ X be ano<strong>the</strong>r non-zero positive operator commut<strong>in</strong>g<br />

with S. If ei<strong>the</strong>r S or B is locally quas<strong>in</strong>ilpotent at a non-zero positive vector, <strong>the</strong>n<br />

<strong>the</strong> operators S and B have a common non-trivial closed <strong>in</strong>variant ideal.<br />

PROOF. If <strong>the</strong> operator S is itself locally quas<strong>in</strong>ilpotent at a non-zero positive vector, <strong>the</strong>n<br />

S is a compact-friendly operator that is locally quas<strong>in</strong>ilpotent. So Theorem 46 is applicable<br />

and guarantees <strong>the</strong> existence <strong>of</strong> <strong>the</strong> desired <strong>in</strong>variant subspace.<br />

Assume now that B is locally quas<strong>in</strong>ilpotent at a non-zero positive vector. Without loss<br />

<strong>of</strong> generality we can assume that S is strictly positive (s<strong>in</strong>ce o<strong>the</strong>rwise <strong>the</strong> null ideal Ns<br />

provides at once a common <strong>in</strong>variant subspace). Therefore, by Lemma 53, <strong>the</strong> operator S 3<br />

dom<strong>in</strong>ates a non-zero compact positive operator K. Consider now <strong>the</strong> operator B 4- S 4- S 3 .<br />

Clearly, it dom<strong>in</strong>ates K and commutes with both S and B. It follows that B is compactfriendly<br />

and by hypo<strong>the</strong>sis B is locally quas<strong>in</strong>ilpotent at a non-zero positive vector. Hence,<br />

Theorem 46, applied to B and B 4- S 4- S 3, guarantees <strong>the</strong> existence <strong>of</strong> a common nontrivial<br />

closed <strong>in</strong>variant ideal for <strong>the</strong>se two operators. S<strong>in</strong>ce both S and B are dom<strong>in</strong>ated by<br />

B 4- S + S 3, it follows that this ideal rema<strong>in</strong>s <strong>in</strong>variant under both S and B.<br />

D<br />

COROLLARY 55. Every positive <strong>in</strong>tegral operator that is locally quas<strong>in</strong>ilpotent at a nonzero<br />

positive vector has a non-trivial closed <strong>in</strong>variant ideal.<br />

For <strong>the</strong> classical Lp-spaces (and, as a matter <strong>of</strong> fact, for any <strong>Banach</strong> function space), <strong>the</strong><br />

preced<strong>in</strong>g corollary yields <strong>the</strong> follow<strong>in</strong>g.<br />

COROLLARY 56. Let B : Lp(#) --+ Lp(~) (where # is a-f<strong>in</strong>ite and 1

110 Y.A. Abramovich and C.D. Aliprantis<br />

We conclude this section by discuss<strong>in</strong>g <strong>the</strong> well known And6-Krieger <strong>the</strong>orem and its<br />

generalizations.<br />

DEFINITION 58. A positive operator T :X --+ X on a <strong>Banach</strong> lattice is ideal irreducible,<br />

if T does not have a non-trivial <strong>in</strong>variant closed ideal, i.e., if T (J) ___ J and J is a closed<br />

ideal, <strong>the</strong>n ei<strong>the</strong>r J = 0 or J = E. Similarly, T is band irreducible, if Tdoes not have a<br />

non-trivial <strong>in</strong>variant band.<br />

In 1957, And6 [31 ] proved that every band irreducible compact positive <strong>in</strong>tegral operator<br />

has a strictly positive spectral radius. Twelve years later, <strong>in</strong> 1969, Krieger [85] removed<br />

<strong>the</strong> compactness condition from And6's <strong>the</strong>orem, and <strong>the</strong> remarkable result assert<strong>in</strong>g that<br />

a band irreducible positive <strong>in</strong>tegral operator has a strictly positive spectral radius, is today<br />

known as <strong>the</strong> And6-Krieger <strong>the</strong>orem. It is obvious that this result is an <strong>in</strong>variant subspace<br />

result as soon as it is reformulated as follows: each quas<strong>in</strong>ilpotent positive <strong>in</strong>tegral operator<br />

has a non-trivial <strong>in</strong>variant band.<br />

The pro<strong>of</strong>s presented by And6 and Krieger were not at all transparent and several attempts<br />

were made to clarify <strong>the</strong> <strong>in</strong>tr<strong>in</strong>sic properties beh<strong>in</strong>d <strong>the</strong> structure <strong>of</strong> kernel operators.<br />

In addition to <strong>the</strong>se efforts, many <strong>in</strong>vestigators looked at <strong>the</strong> possibility <strong>of</strong> generaliz<strong>in</strong>g <strong>the</strong><br />

And6-Krieger <strong>the</strong>orem. In spite <strong>of</strong> all <strong>the</strong>se research activities, it was not until somewhat<br />

recently that <strong>the</strong> And6-Krieger <strong>the</strong>orem was understood relatively well and generalized <strong>in</strong><br />

various directions. Several people have contributed to <strong>the</strong> understand<strong>in</strong>g <strong>of</strong> this subject;<br />

among <strong>the</strong>m Caselles [45,46], Grobler [65,66], de Pagter [106], Schaefer [124,120], and<br />

Zaanen [ 144, Section 136].<br />

The early attempts to generalize <strong>the</strong> And6-Krieger <strong>the</strong>orem to operators more general<br />

than <strong>in</strong>tegral were fraught with many obstacles from <strong>the</strong> outset. Schaefer [ 120] constructed<br />

an example <strong>of</strong> an ideal irreducible operator on Lp(#) (1 < p < e~) which was quas<strong>in</strong>ilpotent.<br />

This showed that even for <strong>the</strong> class <strong>of</strong> ideal irreducible operators (which is larger<br />

than that <strong>of</strong> band irreducible operators) some extra condition is necessary. For a long time it<br />

was not clear whe<strong>the</strong>r <strong>the</strong> compactness would be such a condition. This problem was solved<br />

by de Pagter [ 106] who proved that every ideal irreducible compact positive operator on<br />

a <strong>Banach</strong> lattice has a strictly positive spectral radius. This remarkable <strong>the</strong>orem was <strong>the</strong><br />

start<strong>in</strong>g po<strong>in</strong>t <strong>in</strong> [5], where, <strong>in</strong>stead <strong>of</strong> impos<strong>in</strong>g all conditions on one operator, some <strong>of</strong><br />

<strong>the</strong> properties were shifted to <strong>the</strong> commutant <strong>of</strong> <strong>the</strong> operator. The next two results not only<br />

demonstrate <strong>the</strong> difference between <strong>the</strong> ideal and band irreducible operators but also are<br />

<strong>the</strong> strongest generalizations <strong>of</strong> <strong>the</strong> And6-Krieger <strong>the</strong>orem for ideal and band irreducible<br />

operators obta<strong>in</strong>ed so far. We refer to [5] for <strong>the</strong>se and many relevant results.<br />

THEOREM 59. If an ideal irreducible positive operator U : X ~ X commutes with a compact<br />

positive operator V : X --+ X, <strong>the</strong>n r(UV) > O, and <strong>in</strong> particular r(U) > 0 and<br />

r(V) > O.<br />

PROOF. We will outl<strong>in</strong>e <strong>the</strong> ma<strong>in</strong> po<strong>in</strong>ts <strong>of</strong> <strong>the</strong> pro<strong>of</strong>. First <strong>of</strong> all, let us show that every<br />

non-zero compact quas<strong>in</strong>ilpotent positive operator T has a non-trivial closed positively<br />

hyper<strong>in</strong>variant ideal J, that is, J is <strong>in</strong>variant under each positive operator <strong>in</strong> <strong>the</strong> commutant<br />

{T}' <strong>of</strong> T.

Positive operators 111<br />

Let F be <strong>the</strong> closed ideal generated by <strong>the</strong> range <strong>of</strong> T. We claim that F is <strong>the</strong> closure<br />

<strong>of</strong> a pr<strong>in</strong>cipal ideal. The compactness <strong>of</strong> T implies that <strong>the</strong> range <strong>of</strong> T is separable. So,<br />

<strong>the</strong>re exists a countable subset {yl, y2 .... } <strong>of</strong> T (X) consist<strong>in</strong>g <strong>of</strong> non-zero vectors that is<br />

norm dense <strong>in</strong> T(X). Consider u -- y~'n~__l 2,I~1'~I, ii which is <strong>in</strong> F. For <strong>the</strong> pr<strong>in</strong>cipal ideal<br />

A, generated by u we clearly have A, _ F. Conversely, <strong>the</strong> <strong>in</strong>clusion {yl, y2 .... } _ A,<br />

implies that T (X) ___ A,. Hence, F c_ Au and so A, = F.<br />

Clearly, F 5~ {0} and it is easy to verify that F is <strong>in</strong>deed a positively hyper<strong>in</strong>variant<br />

ideal for T. We are done if F is proper. So assume that F -- A, = X, which implies <strong>in</strong><br />

particular that X has quasi-<strong>in</strong>terior po<strong>in</strong>ts. In this case, <strong>the</strong> existence pro<strong>of</strong> <strong>of</strong> a non-trivial<br />

closed positively hyper<strong>in</strong>variant ideal is just <strong>the</strong> second part <strong>of</strong> <strong>the</strong> pro<strong>of</strong> <strong>of</strong> de Pagter's<br />

result mentioned above [106, Proposition 2]. The basic steps <strong>of</strong> this pro<strong>of</strong> follow.<br />

Let J+ = {0~< S E s 3 R E {T}'with0~< S ~< R}. Also, consider <strong>the</strong> vector<br />

subspace ,7 = {S1 - $2: Sl, $2 E ,.7 + } <strong>of</strong> s In addition, for each x >~ 0, let<br />

J[x] = {Sx: S E J}. It is shown <strong>in</strong> [106] that:<br />

(1) For each x > 0 <strong>the</strong> closure J[x] is a non-zero T-<strong>in</strong>variant closed ideal; and<br />

(2) For some x > 0 <strong>the</strong> closed ideal J[x] is non-trivial.<br />

Next, let C be a positive operator that commutes with T. If S E J, <strong>the</strong>n <strong>the</strong>re exist<br />

S1,82 E ,J+ and RI, R2 E {T}' satisfy<strong>in</strong>g S = Sl - 82 and 0 ~< Si ~ Ri (i -- 1,2). Then,<br />

0

112 Y.A. Abramovich and C.D. Aliprantis<br />

and details. Each <strong>of</strong> <strong>the</strong>se applications requires <strong>the</strong> development <strong>of</strong> fairly sophisticated<br />

mach<strong>in</strong>ery that cannot be presented here because <strong>of</strong> space limitations. More areas <strong>of</strong> applications<br />

can be found <strong>in</strong> <strong>the</strong> <strong>in</strong>terest<strong>in</strong>g surveys by Dancer [49] (devoted to some recent<br />

work on maps, both l<strong>in</strong>ear and nonl<strong>in</strong>ear, between cones <strong>in</strong> <strong>Banach</strong> spaces) and by Nussbaum<br />

[103,104] (devoted to <strong>the</strong> fixed po<strong>in</strong>t <strong>in</strong>dex and some <strong>of</strong> its applications).<br />

Biology. We beg<strong>in</strong> with some <strong>in</strong>terest<strong>in</strong>g and surpris<strong>in</strong>g applications <strong>of</strong> <strong>the</strong> abstract resuits<br />

on dom<strong>in</strong>ation described <strong>in</strong> Section 4. The first example <strong>of</strong> such an application was<br />

discovered by Btirger [42] <strong>in</strong> biology. In [42] a very general discrete-time model that describes<br />

<strong>the</strong> evolution <strong>of</strong> type densities under <strong>the</strong> action <strong>of</strong> mutation and selection <strong>in</strong> an<br />

asexually reproduc<strong>in</strong>g population is developed and analyzed. As a result <strong>of</strong> this analysis,<br />

<strong>the</strong> author proves <strong>the</strong> existence and uniqueness <strong>of</strong> a strictly positive equilibrium.<br />

The analysis unfolds <strong>in</strong> some L1 (#)-space, that is, <strong>in</strong> a special <strong>Banach</strong> lattice with order<br />

cont<strong>in</strong>uous norm and so <strong>the</strong> Aliprantis-Burk<strong>in</strong>shaw Theorem 21 is applicable. This<br />

allows <strong>the</strong> author to conclude that for some positive compactly dom<strong>in</strong>ated <strong>in</strong>tegral operator<br />

U its second power U 2 is compact and <strong>the</strong>n to apply to U <strong>the</strong> improvement <strong>of</strong><br />

<strong>the</strong> Kre<strong>in</strong>-Rutman <strong>the</strong>orem mentioned after Theorem 33, accord<strong>in</strong>g to which r(U) > 0 is<br />

an eigenvalue, and U has a unique strictly positive normalized eigenvector correspond<strong>in</strong>g<br />

to r(U). This model and its analysis have been recently generalized by BUrger and<br />

Bomze [43].<br />

A similar application, but for a completely different model, has been obta<strong>in</strong>ed by<br />

Rhandi [ 113]. He has also applied Theorem 21 <strong>in</strong> his <strong>in</strong>vestigation <strong>of</strong> a population equation<br />

to obta<strong>in</strong> compactness <strong>of</strong> <strong>the</strong> rema<strong>in</strong>der <strong>of</strong> order 3 <strong>of</strong> <strong>the</strong> Dyson-Philips expansion. In his<br />

recent work with Schnaubelt [114], <strong>the</strong>y deal with a more complicated model describ<strong>in</strong>g<br />

a non-autonomous population equation with diffusion <strong>in</strong> L1, and aga<strong>in</strong> apply Theorem 21<br />

to deduce <strong>the</strong> compactness <strong>of</strong> some operators. This allows <strong>the</strong>m to obta<strong>in</strong> existence and<br />

uniqueness <strong>of</strong> positive solutions to this equation.<br />

A beautiful application <strong>of</strong> <strong>the</strong> Perron-Frobenius Theorem 32 to simple growth processes<br />

is given next. Suppose that <strong>the</strong>re are N different types <strong>of</strong> biological objects. Assume also<br />

that at <strong>the</strong> times t = 0, 1, 2 .... each <strong>in</strong>dividual object produces a certa<strong>in</strong> number <strong>of</strong> <strong>of</strong>fspr<strong>in</strong>g<br />

among which every type is present. To put it ma<strong>the</strong>matically, we denote by aij <strong>the</strong><br />

number <strong>of</strong> <strong>of</strong>fspr<strong>in</strong>g <strong>of</strong> type i produced by a s<strong>in</strong>gle object <strong>of</strong> type j.<br />

If we denote by xi(n) <strong>the</strong> number <strong>of</strong> <strong>the</strong> ith type present at time t = n, <strong>the</strong>n <strong>the</strong> size<br />

<strong>of</strong> <strong>the</strong> system at this time is determ<strong>in</strong>ed by N numbers xl (n) ..... xN(n). Hence, <strong>the</strong> size<br />

<strong>of</strong> <strong>the</strong> i th type at time t - n + 1 will be given by Y~Y-1 aijxj (n). In vector form we have<br />

x (n + 1) = Ax (n), where A = [aij ] and <strong>the</strong> vector x (n) = (x 1 (n) ..... XN (n)). We are<br />

<strong>in</strong>terested <strong>in</strong> <strong>the</strong> asymptotics <strong>of</strong> x (n) as time n tends to <strong>in</strong>f<strong>in</strong>ity, that is, we are <strong>in</strong>terested <strong>in</strong><br />

<strong>the</strong> behavior <strong>of</strong> <strong>the</strong> sequence A n (x (0)), where x(0) is <strong>the</strong> <strong>in</strong>itial distribution <strong>of</strong> <strong>the</strong> system.<br />

The Perron-Frobenius Theorem allows us to solve this problem.<br />

Indeed, as we mentioned after Theorem 32, <strong>the</strong> sequence {[r(a)]~<br />

1 A nu}, where u is an<br />

arbitrary positive vector, converges to a unique (up to a multiplier) vector that is coll<strong>in</strong>ear<br />

with <strong>the</strong> unit eigenvector v correspond<strong>in</strong>g to <strong>the</strong> eigenvalue r(A) > 0. This multiplier depends<br />

on our choice <strong>of</strong> <strong>the</strong> vector u. In o<strong>the</strong>r words, <strong>the</strong> sequence {x(n)} asymptotically<br />

behaves as [r (A)]n v.

Positive operators 113<br />

Economics: Leontief's model. A very similar application <strong>of</strong> <strong>the</strong> Perron-Frobenius Theorem<br />

32 to economics is produced by <strong>the</strong> Leontief model. Assume that we have N factories<br />

each <strong>of</strong> which produces a s<strong>in</strong>gle commodity (or good). To produce each commodity o<strong>the</strong>r<br />

commodities must be employed. We denote by aij <strong>the</strong> amount <strong>of</strong> <strong>the</strong> j th good that is<br />

needed to produce one unit <strong>of</strong> <strong>the</strong> i th good. These numbers are called <strong>the</strong> <strong>in</strong>put coefficients<br />

and <strong>the</strong>y are usually assumed to be constant and nonnegative.<br />

Let us denote by di <strong>the</strong> demand for <strong>the</strong> i th good and by xi <strong>the</strong> total output <strong>of</strong> <strong>the</strong> i th good<br />

over a fixed period. Then <strong>the</strong> standard equilibrium equation "Supply -- Demand" yields <strong>the</strong><br />

follow<strong>in</strong>g system <strong>of</strong> equations:<br />

N<br />

Xi -- Z<br />

j-1<br />

aijxj -- di,<br />

i--1 ..... N,<br />

or (I - A)x -- d. It is clear from <strong>the</strong> setup that A >~ 0 and also that <strong>the</strong> <strong>of</strong>f-diagonal entries<br />

<strong>of</strong> I - A are nonpositive numbers. The economic situation is "feasible" if and only if I - A<br />

is nons<strong>in</strong>gular and r(A) ~< 1. Under <strong>the</strong>se conditions <strong>the</strong> system can be uniquely solved for<br />

each output vector d ~> 0 and x - (I - A)-ld ~> 0. For details and various generalizations<br />

see [33,35].<br />

Economics: general equilibrium. Here we shall describe <strong>the</strong> basic framework <strong>of</strong> general<br />

equilibrium <strong>the</strong>ory <strong>in</strong> economics" for details see [23]. The pr<strong>in</strong>cipal ma<strong>the</strong>matical tools<br />

needed <strong>in</strong> this area are functional analysis and positive functionals ra<strong>the</strong>r than operators.<br />

Never<strong>the</strong>less, it seems very appropriate to mention this area <strong>in</strong> order to demonstrate <strong>the</strong><br />

extent to which ma<strong>the</strong>matical sophistication has penetrated work <strong>in</strong> economics and f<strong>in</strong>ance.<br />

The economic <strong>in</strong>tuition <strong>of</strong> commodities and prices is understood by means <strong>of</strong> a dual<br />

system (X, X'). The vector space X is <strong>the</strong> commodity space and X I is <strong>the</strong> price space. The<br />

evaluation (x, x I) is <strong>in</strong>terpreted as <strong>the</strong> value <strong>of</strong> <strong>the</strong> "bundle" x at prices x ~. For simplicity,<br />

we shall assume here that X is a (f<strong>in</strong>ite- or <strong>in</strong>f<strong>in</strong>ite-dimensional) <strong>Banach</strong> lattice and that X ~<br />

is its norm dual.<br />

An exchange economy consists <strong>of</strong> a f<strong>in</strong>ite number <strong>of</strong> consumers, say m, <strong>in</strong>dexed by i<br />

(i -- 1 ..... m) whose commodity-price duality is described by (X, X'). Every consumer i<br />

has a preference relation >-i, i.e., a b<strong>in</strong>ary relation on X+, which allows <strong>the</strong> consumer to<br />

dist<strong>in</strong>guish out <strong>of</strong> any two bundles x and y <strong>the</strong> one which is better for him. The relation<br />

x ___i Y is read "<strong>the</strong> bundle x is preferred to y "The preference ___.i is assumed to be complete<br />

(i.e., any two vectors <strong>in</strong> X+ are comparable), transitive, convex (i.e., <strong>the</strong> better-than-x set<br />

{y E X+" y ___i X } is a convex set for each x E X+), cont<strong>in</strong>uous (i.e., for each x E X+ <strong>the</strong><br />

sets {y E X+" y ~i X} and {y E X+" x ___i Y} are both closed), and monotone (i.e., "more<br />

is better" <strong>in</strong> <strong>the</strong> sense that y > x ~> 0 implies y ___i X). Each consumer i has also an <strong>in</strong>itial<br />

endowment COi > 0. The total or <strong>the</strong> social endowment (or <strong>the</strong> resources) <strong>of</strong> <strong>the</strong> economy<br />

is <strong>the</strong> vector co- Ziml coi.<br />

An allocation (redistribution) is any m-tuple (xl, x2 ..... Xm) such that xi >~ 0 for each i<br />

and Ziml xi - co. The objective <strong>of</strong> <strong>the</strong> economic activity <strong>in</strong> <strong>the</strong> economy is to redistribute<br />

<strong>the</strong> resources among <strong>the</strong> consumers <strong>in</strong> such a way that each consumer becomes better <strong>of</strong>f.<br />

The efficient ways <strong>of</strong> redistribut<strong>in</strong>g <strong>the</strong> goods among <strong>the</strong> consumers are achieved by means

114 Y.A. Abramovich and C.D. Aliprantis<br />

<strong>of</strong> Pareto optimal allocations. An allocation (x l, X2 ..... Xm) is said to be Pareto optimal<br />

if <strong>the</strong>re is no o<strong>the</strong>r allocation (yl, y2 ..... Ym) such that Yi ~-i Xi for each consumer i and<br />

Yi ~i xi for some/(where yi ~i Xi means yi ~i Xi and xi ~i Yi). In o<strong>the</strong>r words, a distribution<br />

<strong>of</strong> <strong>the</strong> resources is Pareto optimal if it is impossible to make some consumer better<br />

<strong>of</strong>f without mak<strong>in</strong>g some o<strong>the</strong>r one worse <strong>of</strong>f. Ano<strong>the</strong>r important property <strong>of</strong> allocations<br />

is that <strong>of</strong> <strong>in</strong>dividual rationality: an allocation (x l, x2 ..... Xm) is <strong>in</strong>dividually rational if<br />

Xi ~i (-t)i holds for each consumer i; that is, <strong>the</strong> redistribution does not give any consumer<br />

a bundle which is worse than his <strong>in</strong>itial endowment. If <strong>the</strong> order <strong>in</strong>terval [0, co] is weakly<br />

compact (<strong>in</strong> particular, if X has order cont<strong>in</strong>uous norm), <strong>the</strong>n an easy application <strong>of</strong> Zorn's<br />

lemma guarantees that Pareto optimal <strong>in</strong>dividually rational allocations exist.<br />

The above efficiency notion requires that <strong>the</strong> acceptable redistributions <strong>of</strong> <strong>the</strong> goods are<br />

Pareto optimal and <strong>in</strong>dividually rational. This br<strong>in</strong>gs us to <strong>the</strong> important decentralization<br />

problem <strong>in</strong> economics, which <strong>in</strong> ma<strong>the</strong>matical term<strong>in</strong>ology can be stated as follows: Is<br />

<strong>the</strong>re a price system (i.e., some l<strong>in</strong>ear functional x ~ E X ~) that can be used to enforce a<br />

given Pareto efficient allocation? Put ano<strong>the</strong>r way: Can we use a system <strong>of</strong> prices - like <strong>the</strong><br />

one we see <strong>in</strong> every day life - to redistribute <strong>the</strong> resources <strong>of</strong> <strong>the</strong> economy?<br />

This problem is related to <strong>the</strong> ma<strong>the</strong>matical notion <strong>of</strong> supportability <strong>of</strong> convex sets by<br />

cont<strong>in</strong>uous l<strong>in</strong>ear functionals. Recall that a subset A <strong>of</strong> a vector space is said to be supported<br />

by a non-zero l<strong>in</strong>ear functional f at some po<strong>in</strong>t a E A if f (a) ~< f (x) holds for all<br />

x 6 A. An allocation (x l, x2 ..... Xm) is said to be supported by a non-zero price x ~ E X ~<br />

if x' supports <strong>the</strong> better-than-x/set <strong>of</strong> each consumer i (i.e., <strong>the</strong> set {x E X+: x ~i Xi }) at<br />

<strong>the</strong> po<strong>in</strong>t xi. That is, x ~ E X~supports <strong>the</strong> allocation (Xl, x2 ..... Xm) if and only if<br />

x • xi---> (x, x') >>, (xi, x').<br />

Now <strong>the</strong> economic concept <strong>of</strong> <strong>the</strong> decentralization <strong>of</strong> <strong>the</strong> allocations f<strong>in</strong>ds its best <strong>in</strong>terpretation<br />

<strong>in</strong> <strong>the</strong> two famous <strong>the</strong>orems <strong>of</strong> welfare economics. These <strong>the</strong>orems were first proved<br />

for f<strong>in</strong>ite dimensional commodity spaces by K. Arrow and G. Debreu and are stated next.<br />

1 ST WELFARE THEOREM. Every allocation supported by prices is Pareto optimal.<br />

2ND WELFARE THEOREM. Every Pareto optimal allocation can be supported by prices.<br />

In particular, <strong>the</strong> second welfare <strong>the</strong>orem is <strong>of</strong> fundamental importance and its validity is<br />

a "must" for any economic system. To establish <strong>the</strong> validity <strong>of</strong> this <strong>the</strong>orem is, <strong>in</strong> general,<br />

a difficult problem, especially when one deals with an <strong>in</strong>f<strong>in</strong>ite dimensional commodity<br />

space. Support<strong>in</strong>g a convex set at a given (boundary) po<strong>in</strong>t by a non-zero cont<strong>in</strong>uous l<strong>in</strong>ear<br />

functional is not always feasible and support<strong>in</strong>g m convex sets each at a specified po<strong>in</strong>t by<br />

<strong>the</strong> same non-zero cont<strong>in</strong>uous l<strong>in</strong>ear functional is, if not impossible, a very serious matter.<br />

To achieve this goal, we must impose some extra conditions.<br />

The existence <strong>of</strong> a support<strong>in</strong>g price for a given Pareto optimal allocation (x 1, x2 ..... Xm)<br />

is proved by utiliz<strong>in</strong>g <strong>the</strong> lattice structure <strong>of</strong> <strong>the</strong> price space X ~. It usually consists <strong>of</strong> two<br />

that supports <strong>the</strong><br />

steps. In <strong>the</strong> first step one proves <strong>the</strong> existence <strong>of</strong> a non-zero price x i

Positive operators 115<br />

better-than-x/ set at Xi. Once this is done, <strong>the</strong>n <strong>the</strong> second step requires a sophisticated<br />

argument to establish that <strong>the</strong> supremum l<strong>in</strong>ear functional<br />

x' -- x' l v x~ v... vx~<br />

is non-zero and supports <strong>the</strong> allocation (x l, x2 . . . . . Xm). The lattice structure <strong>of</strong> X' is <strong>of</strong><br />

fundamental importance. As a matter <strong>of</strong> fact, if X' is not a vector lattice, <strong>the</strong>re are examples<br />

<strong>of</strong> Pareto optimal allocations that cannot be supported by prices.<br />

F<strong>in</strong>ance: choos<strong>in</strong>g <strong>the</strong> optimal portfolio. We shall discuss here how positive operators<br />

appear <strong>in</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> f<strong>in</strong>ance <strong>in</strong> <strong>the</strong> context <strong>of</strong> choos<strong>in</strong>g an optimal portfolio under uncerta<strong>in</strong>ty.<br />

For a systematic presentation <strong>the</strong> reader can consult <strong>the</strong> excellent book [95].<br />

All <strong>in</strong>vestment decisions are made under uncerta<strong>in</strong>ty regard<strong>in</strong>g future f<strong>in</strong>ancial conditions.<br />

To analyze <strong>the</strong> decision process under uncerta<strong>in</strong>ty <strong>in</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> f<strong>in</strong>ance, one usually<br />

<strong>in</strong>troduces a two-period model (<strong>the</strong> simplest possible model), where at period 0 (today)<br />

everyth<strong>in</strong>g is known and at period 1 (tomorrow) everyth<strong>in</strong>g is unknown and uncerta<strong>in</strong>. To<br />

capture <strong>the</strong> notion <strong>of</strong> uncerta<strong>in</strong>ty tomorrow one usually <strong>in</strong>troduces <strong>the</strong> set S <strong>of</strong> all possible<br />

states <strong>of</strong> <strong>the</strong> world tomorrow. The usual assumptions about S is that it is ei<strong>the</strong>r a f<strong>in</strong>ite set<br />

S -- { 1, 2 ..... S }, or a more general compact Hausdorff topological space, or a probability<br />

space (S, P).<br />

By <strong>the</strong> term security one should understand any object or property that can be traded<br />

<strong>in</strong> <strong>the</strong> markets and that has a value today. While any such object (security) has a specific<br />

value today, its value tomorrow is uncerta<strong>in</strong> and depends, <strong>in</strong> general, upon <strong>the</strong> prevail<strong>in</strong>g<br />

state <strong>of</strong> <strong>the</strong> world tomorrow. Thus, a security is characterized by a value (or a pay<strong>of</strong>f)<br />

at each possible state <strong>of</strong> <strong>the</strong> world tomorrow. Ma<strong>the</strong>matically, this naturally leads to <strong>the</strong><br />

<strong>in</strong>terpretation <strong>of</strong> a security simply as a function x : S --+ R, where x(s) is <strong>in</strong>terpreted as <strong>the</strong><br />

value <strong>of</strong> <strong>the</strong> security x tomorrow when <strong>the</strong> state <strong>of</strong> <strong>the</strong> world s has been realized. Obvious<br />

economic considerations require that <strong>the</strong> collection <strong>of</strong> all securities X is a vector space<br />

under <strong>the</strong> usual po<strong>in</strong>twise operations. This space is called <strong>the</strong> space <strong>of</strong> securities. The most<br />

common spaces <strong>of</strong> securities used <strong>in</strong> f<strong>in</strong>ance are C (S) and L2 (S, P); <strong>the</strong> spaces L p (S, P)<br />

also are used quite <strong>of</strong>ten. In sum: <strong>the</strong> vector space <strong>of</strong> securities represents every object or<br />

property available that can be possibly traded <strong>in</strong> <strong>the</strong> market today.<br />

However, <strong>in</strong> today's market not every security is available for trade. For <strong>in</strong>stance, every<br />

house is a security, but not everyone's house is <strong>in</strong> <strong>the</strong> market for sale today. The collection<br />

M <strong>of</strong> all securities that are available for trad<strong>in</strong>g today is assumed to be a vector subspace<br />

<strong>of</strong> X, and this space M is called <strong>the</strong> marketed space. If M -- X, <strong>the</strong>n <strong>the</strong> markets are called<br />

complete and if M -r X <strong>the</strong> markets are <strong>in</strong>complete. From <strong>the</strong> practical po<strong>in</strong>t <strong>of</strong> view <strong>the</strong><br />

markets are always <strong>in</strong>complete. Clearly, <strong>the</strong> structure <strong>of</strong> M should play an important role<br />

<strong>in</strong> mak<strong>in</strong>g decisions today given <strong>the</strong> uncerta<strong>in</strong>ty <strong>of</strong> tomorrow. Now let us discuss <strong>the</strong> hedg<strong>in</strong>g<br />

problem. In order to protect <strong>the</strong>ir securities, given <strong>the</strong> uncerta<strong>in</strong>ty <strong>of</strong> tomorrow, people<br />

are forced to buy <strong>in</strong>surance. An <strong>in</strong>surance is simply a non-negative marketed security. If<br />

a person owns a security x and wishes <strong>the</strong> value <strong>of</strong> this security tomorrow never to fall<br />

below a certa<strong>in</strong> level k (called <strong>the</strong> strik<strong>in</strong>g price or <strong>the</strong> floor), <strong>the</strong>n he must purchase some<br />

<strong>in</strong>surance a c M+ = M A X+ so that x + a ~> k, or x(s) + a(s) >7 k for each state s <strong>of</strong> <strong>the</strong><br />

world tomorrow. This implies a ~ k - x, and hence a ~> (k - x) +, where (k - x)+(s) =

116 YA. Abramovich and C.D. Aliprantis<br />

max{k - x(s), 0}. Therefore, <strong>the</strong> "smallest" desired <strong>in</strong>surance seems to be (k - x) + and it<br />

is called <strong>the</strong> put option <strong>of</strong> <strong>the</strong> security x with strik<strong>in</strong>g price k. S<strong>in</strong>ce M need not be a vector<br />

sublattice <strong>of</strong> X, <strong>the</strong> put option (k - x) + need not be available <strong>in</strong> <strong>the</strong> market today, i.e.,<br />

(k - x) + need not belong to M. This is why <strong>the</strong> <strong>the</strong>ory <strong>of</strong> lattice-subspaces plays an important<br />

role here along with <strong>the</strong> <strong>the</strong>ory <strong>of</strong> <strong>Banach</strong> and vector lattices (see for <strong>in</strong>stance [ 14,24]).<br />

And now we can state <strong>the</strong> hedg<strong>in</strong>g problem <strong>of</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> f<strong>in</strong>ance.<br />

THE HEDGING PROBLEM. Given a marketed security x, what is <strong>the</strong> "cheapest <strong>in</strong>surance"<br />

that one can purchase <strong>in</strong> order to guarantee at least <strong>the</strong> value k for <strong>the</strong> security<br />

tomorrow no matter what <strong>the</strong> prevail<strong>in</strong>g state will be?<br />

Of course, if M is a vector sublattice <strong>of</strong> X and x, k 6 M, <strong>the</strong>n a = (k - x) + should be <strong>the</strong><br />

desired answer. The general solution <strong>of</strong> <strong>the</strong> Hedg<strong>in</strong>g Problem is non-trivial and is related<br />

to <strong>the</strong> mean<strong>in</strong>g <strong>of</strong> <strong>the</strong> "cheapest <strong>in</strong>surance"9 This is <strong>the</strong> place where positive operators and<br />

<strong>the</strong>ir properties make <strong>the</strong>ir appearance.<br />

To simplify matters fur<strong>the</strong>r, let us assume that <strong>the</strong>re is ei<strong>the</strong>r a f<strong>in</strong>ite or a countable<br />

number <strong>of</strong> l<strong>in</strong>early <strong>in</strong>dependent securities whose l<strong>in</strong>ear span is M. Consider<strong>in</strong>g <strong>the</strong> simplest<br />

case <strong>of</strong> <strong>the</strong> two, we will suppose that <strong>the</strong>re are N securities x l, x2 ..... XN <strong>in</strong> M+ that span<br />

M and that S = { 1, 2 ..... S}. In this case we have N O}.

Positive operators 117<br />

Then a price p 6 M* is arbitrage free if and only if (p, RO) = (R* p, 0) > 0 for each 0 6 K,<br />

where R* :M* --+ ~I~ u is <strong>the</strong> adjo<strong>in</strong>t operator <strong>of</strong> R. This duality property is very useful <strong>in</strong><br />

determ<strong>in</strong><strong>in</strong>g <strong>the</strong> arbitrage free prices. From economic considerations only arbitrage free<br />

prices are acceptable prices and <strong>the</strong> Hedg<strong>in</strong>g Problem can be stated now as follows: Given<br />

an arbitrage free price p 6 M*, a security x and a strik<strong>in</strong>g price k, f<strong>in</strong>d <strong>the</strong> portfolio 0 E R x<br />

that m<strong>in</strong>imizes <strong>the</strong> expression (R* p, O) = (p, RO) subject to RO ~ x x/ k.<br />

The Hedg<strong>in</strong>g Problem is also related to <strong>the</strong> notion <strong>of</strong> a derivative <strong>in</strong> <strong>the</strong> <strong>the</strong>ory <strong>of</strong> f<strong>in</strong>ance<br />

and has been studied <strong>in</strong> many contexts. A derivative is a security <strong>of</strong> <strong>the</strong> form<br />

x(s) = f(yl(s), y2(s) ..... yk(s)), where (<strong>in</strong> <strong>the</strong> case X = C(S)) f :R k --+ R is a cont<strong>in</strong>uous<br />

function and yl, y2 ..... yk are k given securities; that is a derivative is a security<br />

whose pay<strong>of</strong>f depends on <strong>the</strong> pay<strong>of</strong>fs <strong>of</strong> a f<strong>in</strong>ite number <strong>of</strong> o<strong>the</strong>r securities.<br />

Acknowledgment<br />

The authors would like to express <strong>the</strong>ir thanks to A. Kitover and V. Troitsky for many<br />

useful suggestions.<br />

Added <strong>in</strong> Pro<strong>of</strong><br />

Detailed pro<strong>of</strong>s <strong>of</strong> <strong>the</strong> results discussed <strong>in</strong> this survey as well as an extensive treatment <strong>of</strong><br />

many related topics can be found <strong>in</strong> <strong>the</strong> forthcom<strong>in</strong>g book "An <strong>in</strong>vitation to Operator Theory"<br />

by <strong>the</strong> authors that will be published shortly by <strong>the</strong> American Ma<strong>the</strong>matical Society.<br />

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13 (3) (1979), 81-82.

CHAPTER 3<br />

Lp <strong>Spaces</strong><br />

Dale Alspach<br />

Department <strong>of</strong> Ma<strong>the</strong>matics, Oklahoma State University, Stillwater, OK, USA<br />

E-mail: alspach @ math. okstate, edu<br />

Edward Odell*<br />

Department <strong>of</strong> Ma<strong>the</strong>matics, The University <strong>of</strong> Texas, Aust<strong>in</strong>, TX, USA<br />

E-mail: odell@ math. utexas, edu<br />

Contents<br />

1. Prelim<strong>in</strong>aries .................................................. 126<br />

2. Global structure ................................................. 129<br />

3. Sequences <strong>in</strong> ~ p and L p ............................................ 134<br />

4. Subspaces <strong>of</strong> L p ................................................ 140<br />

5. /2p-spaces, 1 < p < cx~, p-f: 2 ......................................... 146<br />

References ..................................................... 156<br />

*Research supported by NSE<br />

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 1<br />

Edited by William B. Johnson and Joram L<strong>in</strong>denstrauss<br />

9 2001 Elsevier Science B.V. All rights reserved<br />

123

L p spaces 125<br />

In this chapter we will discuss <strong>the</strong> structure <strong>of</strong> <strong>the</strong> L p-spaces and <strong>the</strong>ir subspaces.<br />

We are concerned ma<strong>in</strong>ly with <strong>the</strong> reflexive case (1 < p < ~). The space L~ and <strong>the</strong><br />

/21-spaces are considered elsewhere [90]. A <strong>the</strong>ory requires and feeds on its examples. The<br />

L p-spaces have provided much fodder for <strong>the</strong> general <strong>the</strong>ory <strong>of</strong> <strong>Banach</strong> spaces because<br />

<strong>the</strong>y appeared early <strong>in</strong> <strong>the</strong> <strong>the</strong>ory and <strong>the</strong> study <strong>of</strong> <strong>the</strong>se spaces has motivated <strong>the</strong> def<strong>in</strong>itions<br />

<strong>of</strong> many properties <strong>of</strong> more general <strong>Banach</strong> spaces. For example, with its usual norm L p<br />

is a <strong>Banach</strong> lattice under <strong>the</strong> po<strong>in</strong>twise almost everywhere order<strong>in</strong>g. In <strong>the</strong> reflexive case it<br />

also has an unconditional basis and thus has ano<strong>the</strong>r lattice structure as a sequence space<br />

(and is a <strong>Banach</strong> lattice under a different norm except <strong>in</strong> <strong>the</strong> case p - 2). These spaces<br />

naturally occur as <strong>in</strong>terpolation spaces and are <strong>the</strong> simplest <strong>of</strong> <strong>the</strong> rearrangement <strong>in</strong>variant<br />

spaces. The Hardy spaces, Hp(D), <strong>of</strong> analytic functions on <strong>the</strong> unit disk with Lp boundary<br />

values are isomorphic to Lp, 1 < p < cxz, and <strong>the</strong> Bergman spaces Ap(D) <strong>of</strong> analytic<br />

functions on <strong>the</strong> unit disk which are <strong>in</strong> Lp(D) are isomorphic to ~p, 1 ~< p < e~, [67]. The<br />

study <strong>of</strong> <strong>the</strong> structure <strong>of</strong> <strong>the</strong> f<strong>in</strong>ite dimensional subspaces <strong>of</strong> L p paved <strong>the</strong> way for much<br />

<strong>of</strong> <strong>the</strong> extraord<strong>in</strong>ary development <strong>of</strong> <strong>the</strong> local <strong>the</strong>ory <strong>of</strong> <strong>Banach</strong> spaces <strong>in</strong> <strong>the</strong> 1980's [77].<br />

In <strong>in</strong>vestigations <strong>of</strong> o<strong>the</strong>r <strong>Banach</strong> spaces and operators <strong>the</strong> existence and classification <strong>of</strong><br />

operators from, <strong>in</strong>to or factor<strong>in</strong>g through L p-spaces provide fundamental <strong>in</strong>formation on<br />

<strong>the</strong> structure.<br />

We will concern ourselves primarily with <strong>the</strong> <strong>in</strong>f<strong>in</strong>ite dimensional isomorphic structure<br />

<strong>of</strong> <strong>the</strong> Lp-spaces. In particular we shall concentrate on separable Lp-spaces and, as noted<br />

<strong>in</strong> <strong>the</strong> basic concepts chapter, [49, Section 4], this reduces essentially to study<strong>in</strong>g g p and<br />

L p(O, 1) (which we shall denote by L p). Functions are assumed to be real-valued as <strong>in</strong><br />

general <strong>the</strong>re are only m<strong>in</strong>or adjustments needed for <strong>the</strong> complex case.<br />

As <strong>the</strong> <strong>the</strong>ory <strong>of</strong> Lp-spaces was developed some natural questions arose. What are <strong>the</strong><br />

complemented subspaces <strong>of</strong> Lp (or ~p).9 When does a given <strong>Banach</strong> space X embed <strong>in</strong>to<br />

Lp (or gp)? If X c_ Lp, what subspaces must live <strong>in</strong>side <strong>of</strong> X? Does every <strong>Banach</strong> space<br />

conta<strong>in</strong> some g p or co? What can be said about <strong>the</strong> structure <strong>of</strong> an unconditional basic<br />

sequence <strong>in</strong> L p ? We address <strong>the</strong>se problems and o<strong>the</strong>rs below. With apologies to <strong>the</strong> experts<br />

we choose to <strong>in</strong>clude some <strong>of</strong> <strong>the</strong> well known (to <strong>the</strong>m) structural results that were<br />

mentioned <strong>in</strong> <strong>the</strong> basic concepts chapter or appear <strong>in</strong> books such as [70].<br />

In Section 1 we review certa<strong>in</strong> <strong>in</strong>equalities for sequences <strong>in</strong> L p. For example we show<br />

that by <strong>in</strong>tegrat<strong>in</strong>g aga<strong>in</strong>st <strong>the</strong> Rademacher functions one can deduce that a normalized<br />

unconditional basic sequence <strong>in</strong> Lp (2 < p < CX~) admits upper ~2 and lower gp estimates<br />

(respectively, for 1 < p < 2, upper gp and lower ~2 estimates). In Section 2 we study <strong>the</strong><br />

global structure <strong>of</strong> L p and <strong>in</strong> particular <strong>the</strong> Haar basis. Among <strong>the</strong> results given we exam<strong>in</strong>e<br />

<strong>the</strong> span <strong>of</strong> subsequences <strong>of</strong> <strong>the</strong> Haar basis and Schechtman's result that every complemented<br />

subspace <strong>of</strong> L p with an unconditional basis is isomorphic to a complemented<br />

subspace spanned by a block basis <strong>of</strong> <strong>the</strong> Haar basis.<br />

Section 3 beg<strong>in</strong>s with some properties and characterizations <strong>of</strong> <strong>the</strong> unit vector basis <strong>of</strong><br />

p. The details <strong>of</strong> <strong>the</strong> argument us<strong>in</strong>g <strong>the</strong> Pelczyfiski decomposition method that a complemented<br />

subspace <strong>of</strong> g p is isomorphic to g p are presented to help give context and mean<strong>in</strong>g<br />

to <strong>the</strong> notion <strong>of</strong> (p, 2) bounded operators discussed later <strong>in</strong> Section 5. We also discuss<br />

spread<strong>in</strong>g models and types and <strong>the</strong> Kriv<strong>in</strong>e-Maurey <strong>the</strong>orem. This latter result gives a<br />

sufficient condition, <strong>in</strong> terms <strong>of</strong> types, for a <strong>Banach</strong> space X to conta<strong>in</strong> almost isometric<br />

copies <strong>of</strong> g p for some p. Section 4 deals with subspaces X <strong>of</strong> L p. For example if

126 D. Alspach and E. Odell<br />

X C Lp (2 < p < cx~) <strong>the</strong>n X must conta<strong>in</strong> an isomorph <strong>of</strong> s or s and if X does not<br />

conta<strong>in</strong> s it must embed <strong>in</strong>to s We also consider a necessary and sufficient condition for<br />

a reflexive space X to embed <strong>in</strong>to <strong>the</strong> s sum (1 < p < cx~) <strong>of</strong> f<strong>in</strong>ite dimensional spaces.<br />

The last section concerns/2p-spaces- those complemented subspaces <strong>of</strong> L p which are not<br />

Hilbert spaces. Here <strong>the</strong> known examples and <strong>the</strong>ir isomorphic classification are discussed.<br />

In particular Rosenthal's space X p and its generalizations are presented <strong>in</strong> detail. The role<br />

<strong>of</strong> (p, 2)-bounded operators is also expla<strong>in</strong>ed. F<strong>in</strong>ally we discuss some <strong>of</strong> <strong>the</strong> results for<br />

s<br />

with )~ near one and <strong>the</strong>ir relation to <strong>the</strong> isometric <strong>the</strong>ory <strong>of</strong> Lp-spaces.<br />

1. Prelim<strong>in</strong>aries<br />

We first recall a few key properties <strong>of</strong> L p and s<br />

concepts chapter.<br />

The unit vector basis for s<br />

which are discussed throughout <strong>the</strong> basic<br />

is a 1-symmetric basis [49, Section 3]. The Haar basis<br />

(hi)~ is an unconditional basis <strong>of</strong> Lp for 1 < p < c~ [49, Section 3], [24]. It is also a<br />

monotone basis for Lp for 1 ~< p < c~. The Rademacher functions (rn)n~=l , [49, Section<br />

4], are equivalent to <strong>the</strong> unit vector basis <strong>of</strong> s for p < ~, (and <strong>the</strong> unit vector basis <strong>of</strong> el<br />

for p=~).<br />

Thus for 0 < p < oc <strong>the</strong>re exist constants A p, B p with<br />

Ap(~lanl2)l/2 (fo I )l/p 2)1/2<br />

L p spaces 127<br />

if 1 ~< p ~< 2, and<br />

(i011<br />

Ilxillp ~< ~ri(t)xi<br />

1<br />

dt

128 D. Alspach and E. Odell<br />

(1.2) and (1.3) can be viewed as generalizations <strong>of</strong> Clarkson's <strong>in</strong>equalities [29]. S<strong>in</strong>ce<br />

II 9 IILp ~< I1" IlL2 for p ~< 2 we also have us<strong>in</strong>g (1.2) for p = 2 that<br />

(So ~ri(t)xi<br />

1<br />

dt ~< IIx/ll 2<br />

p<br />

1/2<br />

for 1 ~ p < 2, (1.5)<br />

and similarly<br />

(So<br />

~-~ ri (t)xi<br />

1<br />

at 1> ilxill 2<br />

p<br />

1/2<br />

for 2 < p. (1.6)<br />

The technique <strong>of</strong> <strong>in</strong>tegrat<strong>in</strong>g aga<strong>in</strong>st <strong>the</strong> Rademacher yields some useful <strong>in</strong>equalities for<br />

unconditional basic sequences <strong>in</strong> L p. If (Xn) is a A-unconditional basic sequence <strong>in</strong> L p<br />

<strong>the</strong>n<br />

[f01 2)p/2 ]l/p<br />

A-I (Elanl2iXn(S)l as I[Ea, x, II"<br />

<strong>in</strong>dependent mean zero random variables <strong>in</strong> L p <strong>the</strong>n<br />

L p spaces 129<br />

max Ilxi IlPp , Ilxi II 2<br />

i:1 i--1<br />

~Xi<br />

t<br />

~ Kpmax Ilxi lip<br />

i:1 p i=1<br />

, IIx/ll 9 (1.11)<br />

It is shown <strong>in</strong> [55] that Kp ~ p~ In p.<br />

A <strong>Banach</strong> space X is a L;p,Z-space if for all f<strong>in</strong>ite dimensional spaces F ___ X <strong>the</strong>re exists<br />

a f<strong>in</strong>ite dimensional E with F c E c X so that d(E fdimE) ~ )~ It ultimately turns out<br />

m ~ vp ,<br />

(see Section 5) that a separable X is s for some )~ and 1 < p < ec iff X is isomorphic<br />

to a complemented subspace <strong>of</strong> L p which is not isomorphic to Hilbert space [66,68].<br />

The situation for L 1 is more complicated. It is conjectured that every <strong>in</strong>f<strong>in</strong>ite dimensional<br />

complemented subspace X <strong>of</strong> L1 is isomorphic to L1 or el. It is known that if X conta<strong>in</strong>s<br />

an isomorph <strong>of</strong> L1 <strong>the</strong>n X is isomorphic to L1 [36] and if X embeds <strong>in</strong>to el <strong>the</strong>n X is<br />

isomorphic to el [65]. Various characterizations <strong>of</strong>/21- (and L;oc-) spaces are given <strong>in</strong> [68].<br />

Much work was done to study and attempt to classify <strong>the</strong> L;p-spaces up to isomorphism<br />

and this is discussed <strong>in</strong> Section 5 below.<br />

We beg<strong>in</strong> with some results on <strong>the</strong> global structure <strong>of</strong> L p and <strong>in</strong> particular those <strong>in</strong>volv<strong>in</strong>g<br />

<strong>the</strong> Haar basis. All <strong>Banach</strong> spaces are presumed to be separable unless o<strong>the</strong>rwise<br />

stated.<br />

2. Global structure<br />

In Hilbert space strong geometric properties follow from <strong>the</strong> properties <strong>of</strong> <strong>the</strong> <strong>in</strong>ner product.<br />

Early <strong>in</strong> <strong>the</strong> development <strong>of</strong> operator <strong>the</strong>ory on more general spaces Murray [81 ] recognized<br />

<strong>the</strong> critical role that <strong>the</strong> orthogonal projection played for operators on L2 and<br />

attempted to f<strong>in</strong>d generalizations for <strong>the</strong> L p-spaces. While we know now (and as Murray<br />

himself discovered <strong>in</strong> part) that <strong>the</strong>re are far fewer bounded projections on <strong>the</strong> typical <strong>Banach</strong><br />

spaces than on Hilbert space, never<strong>the</strong>less, spaces such as L p possess a rich family<br />

<strong>of</strong> projections and <strong>of</strong> correspond<strong>in</strong>g complemented subspaces.<br />

Given <strong>the</strong> unconditionality <strong>of</strong> <strong>the</strong> Haar basis <strong>the</strong> simplest way to get a complemented<br />

subspace <strong>of</strong> Lp (1 < p < oc) is to take [(hji)]~__l where (hji) is a subsequence <strong>of</strong> (hj).<br />

There are however only two isomorphism types <strong>of</strong> such subspaces.<br />

THEOREM 1 ([39]). Let 1 < p < oc and let X -- [(h j;)] for some subsequence (hji) <strong>of</strong><br />

(h j). Then X is isomorphic to e p or L p.<br />

PROOF. By <strong>the</strong> decomposition method [49, Section 4], (or see <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Theorem 15<br />

c c c<br />

below) it suffices to prove that ei<strong>the</strong>r X ~ e p or L p ~ X. (X ~ Y means that X is<br />

isomorphic to a complemented subspace <strong>of</strong> Y.) Let<br />

A = { t c [0, 1 ]" t c supp hji for <strong>in</strong>f<strong>in</strong>itely many i's}.

- {t<br />

- U(An<br />

130 D. Alspach and E. Odell<br />

C<br />

We shall show that X ~ s if m (A) -- 0 and o<strong>the</strong>rwise L p ~ X.<br />

So assume m(A) = 0. We shall produce a sequence (Sn) <strong>of</strong> disjo<strong>in</strong>t measurable subsets<br />

<strong>of</strong> [0, 1] with m ([0, 1] \ Un Sn) = 0 and such that every element <strong>of</strong> X is constant on each<br />

Sn. It <strong>the</strong>n follows that<br />

C<br />

X ~-+ [(l&)]~ g.p.<br />

To do this it suffices to show that if n E N and<br />

An -<br />

~ [0, 1]" t ~ supphji for exactly n i's},<br />

<strong>the</strong>n An may be split <strong>in</strong>to such a sequence <strong>of</strong> sets. Let (Xi) be <strong>the</strong> subsequence <strong>of</strong> those<br />

h ji's <strong>in</strong>volved <strong>in</strong> <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> An and let (yi) be <strong>the</strong> (possibly f<strong>in</strong>ite) subsequence <strong>of</strong><br />

(xi) determ<strong>in</strong>ed by Ui supp yi ~ An and if Xn :/= Yi <strong>the</strong>n suppxn g supp Yi. The yi's are<br />

disjo<strong>in</strong>tly supported and<br />

An -<br />

n supp y+) U U(An n supp y?).<br />

i<br />

i<br />

This gives <strong>the</strong> desired splitt<strong>in</strong>g <strong>of</strong> An.<br />

If m(A) > 0 we partition (hji) <strong>in</strong>to a countable number <strong>of</strong> subsequences (hji)iEN k so<br />

that supp hji A supp h je = 0 if i -J= e 6 Ark and so that if k >~ 2, i ~ Nk <strong>the</strong>n <strong>the</strong>re exists<br />

~ Nk-1 with supphji c_ supphje. Set Bk = UicNk supphji" Clearly<br />

B1 ___ B2 ___ "" and (-]Bk -- A.<br />

One now chooses a subsequence (kn) <strong>of</strong> N with m (Bk, \ A) < en, en .[. 0 rapidly. Def<strong>in</strong>e<br />

do- L<br />

and <strong>in</strong>ductively for m odd<br />

d,- L hji<br />

icNk o i~Nk 1<br />

d2"+m -- Z<br />

hji summed over {i 6 Ark," supphji ___ suppd2n_l+~(m+l)/2]]<br />

+ },<br />

and for m even<br />

d2n+m-Zhji summed over {i ~ Nk," supphji c_ d~_,+~(m+,)/2~}.<br />

A perturbation argument yields that (dn) is equivalent to <strong>the</strong> Haar basis and it is not hard<br />

to see that [(dn)] is complemented <strong>in</strong> X, [10].<br />

D<br />

A vector-valued version <strong>of</strong> this <strong>the</strong>orem exists [78]. There also exists a f<strong>in</strong>ite dimensional<br />

version <strong>of</strong> this <strong>the</strong>orem.

L p spaces 131<br />

THEOREM 2 ([80]). Let 1 < p < cx~. There exists Kp < ~X) so that if (hji)<strong>in</strong>___l is anyf<strong>in</strong>ite<br />

subsequence <strong>of</strong> (hj) and E -- ((hji)~) <strong>the</strong>n d(E, ~np) 1.)<br />

(b) ([79]) Let 1 < p :/= 2 < cx~. There exists Kp >/ 1 and for all n a rearrangement<br />

(hn~2 n 2 n n 2 n<br />

~'i 'i=0 ~ (hi)i=o so that if M E N and (zi) M is a normalized block basis <strong>of</strong> (b i )i=0 <strong>the</strong>n<br />

<strong>the</strong>re exists A c {1 ..... M} with [A[ ~> M/2 so that (Zi)iEA is Kp equivalent to <strong>the</strong> unit<br />

Ial<br />

vector basis <strong>of</strong> g~ p .<br />

The rearrangement is def<strong>in</strong>ed <strong>in</strong> terms <strong>of</strong> <strong>the</strong> supports <strong>of</strong> <strong>the</strong> Haar functions. For i < j<br />

ei<strong>the</strong>r supp b n and supp b n j are disjo<strong>in</strong>twithsuppbi n ly<strong>in</strong>g to <strong>the</strong> left <strong>of</strong> supp bj n orsuppbi n _<br />

suppbj.<br />

Every s has a basis ([54], see Section 5 below) but it is still open as to whe<strong>the</strong>r<br />

or not it must have an unconditional basis. However those that do can be realized isomorphically<br />

as be<strong>in</strong>g spanned by block bases <strong>of</strong> (hn).<br />

THEOREM 4 ([95]). Let 1 < p < cx~ and let X be a complemented subspace <strong>of</strong> Lp with an<br />

unconditional basis (Xn). Then <strong>the</strong>re exists a block basis (Yn) <strong>of</strong> <strong>the</strong> Haar basis with [(Yn)]<br />

complemented <strong>in</strong> L p and such that (Xn) is equivalent to (Yn).<br />

The first part <strong>of</strong> this <strong>the</strong>orem follows from <strong>the</strong> follow<strong>in</strong>g proposition.<br />

PROPOSITION 5 ([95]). Let (Xn) be an unconditional basic sequence <strong>in</strong> Lp (1 < p < ~).<br />

Then <strong>the</strong>re exists an equivalent block basis (Yn) <strong>of</strong> (hn).<br />

PROOF. From (1.7) and (1.8) it follows that if (yi) is ano<strong>the</strong>r unconditional basic sequence<br />

<strong>in</strong> Lp with [yi[ -- Ixi[ <strong>the</strong>n (yi) is equivalent to (xi). By a perturbation argument we may<br />

assume that <strong>the</strong>re exist <strong>in</strong>tegers m n ~ ~ and scalars (ai) so that<br />

2 mn - l<br />

Xn-- Z azmn+klh2mn+k["<br />

k=0<br />

(2.1)<br />

One def<strong>in</strong>es <strong>the</strong> yn'S via this formula by remov<strong>in</strong>g <strong>the</strong> absolute values. [3

fi2<br />

132 D. Alspach and E. Odell<br />

Before complet<strong>in</strong>g <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Theorem 4 we recall a useful isomorphic representation<br />

<strong>of</strong> Lp as Lp(~2) for 1 < p < cxz. The latter is <strong>the</strong> <strong>Banach</strong> space <strong>of</strong> all sequences (fi) <strong>of</strong><br />

measurable functions on [0, 1] with<br />

II(fi)llL,r -<br />

(t) dt 2 it can be shown that (fij) is unconditional and s<strong>in</strong>ce (fi) is<br />

a subsequence <strong>of</strong> (fij) we obta<strong>in</strong> a projection <strong>of</strong> Lp(s onto [(fi)]. (fi) is equivalent<br />

to (xi) and is a block basis <strong>of</strong> (h 2mn+e) ~'2mn--1 1,k=0 Fur<strong>the</strong>rmore (h 2mn+k) is equivalent to<br />

n~<br />

(h2mn+k) and this completes <strong>the</strong> pro<strong>of</strong>for p > 2. For p < 2 a duality argument is necessary<br />

[951. D<br />

Some <strong>Banach</strong> spaces have <strong>the</strong> property that whenever <strong>the</strong>y are isomorphic to a subspace<br />

<strong>of</strong> ano<strong>the</strong>r nice enough <strong>Banach</strong> space that <strong>the</strong>re is ano<strong>the</strong>r isomorphic embedd<strong>in</strong>g with better<br />

properties. One property <strong>of</strong> this type is reproducibility. A basis (xi) for a <strong>Banach</strong> space<br />

X is reproducible if whenever X ___ Y and Y has a basis (yi) <strong>the</strong>n some block basis <strong>of</strong> (yi)<br />

is equivalent to (xi). It is easy to see that because <strong>the</strong>y are weakly null and subsymmetric<br />

that <strong>the</strong> unit vector bases <strong>of</strong> s (1 < p < cxz) or co are reproducible. It is also not hard to<br />

show that <strong>the</strong> unit vector basis <strong>of</strong> s is reproducible.<br />

THEOREM 6 ([67]). The Haar basis for Lp (1 ~< p < cx~) is a reproducible basis. Moreover<br />

<strong>the</strong> same is true for any unconditional basis <strong>of</strong> Lp (1 < p < cx~).<br />

PROOF. By virtue <strong>of</strong> Proposition 5 it suffices to prove that (hi) is a reproducible basis for<br />

L p for 1 ~< p < oc. Let L p c X where X has a basis (Xn) with biorthogonal functionals<br />

(Xn*). By Liapoun<strong>of</strong>f's convexity <strong>the</strong>orem for each f<strong>in</strong>ite set F ___ X* and A ___ [0, 1] with

Lp spaces 133<br />

m(A) > 0 <strong>the</strong>re exists a measurable set B c_ A with m(B) -- m(A \ B) and x*(1B) --<br />

x*(1A\B) for x* E F.<br />

Let en ,~ 0. We can use <strong>the</strong> above to produce a dyadic tree <strong>of</strong> sets (An,j a~'2n +n=0, j= 1 as <strong>in</strong> <strong>the</strong><br />

supports <strong>of</strong> <strong>the</strong> Haar basis and a sequence (fn) as follows. Set f0 -- 1 a0,1 = 1,<br />

fl -- 1Al,l -- 1Al,2, f2 -- 1A2,1 -- 1A2,2<br />

and so on. Then (f,) is 1-equivalent to (hn) and for some subsequence (Pk) <strong>of</strong> N<br />

O

134 D. Alspach and E. Odell<br />

<strong>of</strong> (Xn) for some subsequence (X<strong>in</strong>) and Ixi* n (Txi~)l ~ e for all n and some fixed e > O.<br />

Then (Txi~) is equivalent to (xi~) and [(Tx<strong>in</strong>)] is complemented <strong>in</strong> X.<br />

By <strong>the</strong> lemma (Px(h <strong>in</strong>,jn)) is equivalent to (h <strong>in</strong>'jn) and [Px(h <strong>in</strong>,j~ )] is complemented <strong>in</strong><br />

Lp(~2). Also (h i"'j~) is equivalent to (hi~) and this completes <strong>the</strong> pro<strong>of</strong>.<br />

D<br />

The nonseparable Lp-spaces are built on <strong>the</strong> spaces gp(F) and Lp{O, 1} ~. If one is<br />

consider<strong>in</strong>g only separable subspaces X <strong>of</strong> such spaces <strong>the</strong>n X ~ L p. But <strong>the</strong> global<br />

structure <strong>of</strong> nonseparable L p-spaces can be somewhat different. These spaces need not<br />

have an unconditional (uncountable) basis. Recall dim X = <strong>in</strong>f{ot: <strong>the</strong>re exists S C X with<br />

S dense <strong>in</strong> X and card S = o~}.<br />

THEOREM 9 ([37]). Let 1 < p # 2 < oo and let # be a f<strong>in</strong>ite measure with dimLp(#) ~><br />

l'~co. Then L p (#) does not embed <strong>in</strong>to a space with an unconditional basis.<br />

1<br />

3. Sequences <strong>in</strong> gp and Lp<br />

In this section we look at some special subspaces <strong>of</strong> ~p and L p which are isomorphic to<br />

gr, for some r. Of course for ~p, <strong>the</strong> only possible value <strong>of</strong> r is p itself, but for Lp <strong>the</strong>re<br />

is always r = 2 and for p < 2 an entire <strong>in</strong>terval <strong>of</strong> possibilities. General sequences and<br />

even unconditional basic sequences <strong>in</strong> L p are probably too varied to ever be described (see<br />

Section 10, [50], for a particularly peculiar subspace with unconditional basis), however,<br />

near <strong>the</strong> end <strong>of</strong> this section we note some results for Orlicz sequences and o<strong>the</strong>r spaces<br />

with a symmetric basis. In <strong>the</strong> previous section we looked at <strong>the</strong> span <strong>of</strong> subsequences <strong>of</strong><br />

<strong>the</strong> Haar system. For <strong>the</strong> unit vector basis <strong>of</strong> gp, subsequences are not mysterious, and <strong>in</strong><br />

this simpler situation we can successfully describe <strong>the</strong> span <strong>of</strong> block bases.<br />

Let (ei) be <strong>the</strong> unit vector basis <strong>of</strong> ~p. We beg<strong>in</strong> by recall<strong>in</strong>g <strong>the</strong> a simple fact about<br />

block bases <strong>of</strong> (ei) which was mentioned <strong>in</strong> [49, Section 64], [71, Proposition 2.a. 1 ], [44,<br />

Proposition 1.5.3].<br />

PROPOSITION 10. Suppose that (Xn) is a block basis <strong>of</strong> <strong>the</strong> unit vector basis <strong>of</strong> g.p.<br />

[Xn: n E IN] is isometric to g.p with <strong>the</strong> isometry which is <strong>in</strong>duced by <strong>the</strong> basis map<br />

Xn ----> ]]Xn lien, n = 1, 2 .....<br />

A normalized basic sequence is said to be perfectly homogeneous if it is equivalent to all<br />

<strong>of</strong> its normalized block bases. We have thus shown that <strong>the</strong> unit vector basis <strong>of</strong> g p has this<br />

property. It is easy to see that <strong>the</strong> unit vector basis for co is also perfectly homogeneous. It<br />

turns out that this is a very special property.<br />

THEOREM 11 ([101]). Suppose that X is a <strong>Banach</strong> space with a normalized perfectly<br />

homogeneous basis (Xn), <strong>the</strong>n (Xn) is equivalent to <strong>the</strong> unit vector basis <strong>of</strong> g p or co.<br />

We refer <strong>the</strong> reader to Zipp<strong>in</strong>'s paper or <strong>the</strong> book [71, p. 60], for <strong>the</strong> pro<strong>of</strong>. There have<br />

been some generalizations <strong>of</strong> this result <strong>in</strong> <strong>the</strong> context <strong>of</strong> symmetric spaces [9].

Lp spaces 135<br />

Ano<strong>the</strong>r <strong>in</strong>terest<strong>in</strong>g property <strong>of</strong> block bases <strong>in</strong> ~ p mentioned <strong>in</strong> [49, Section 4] is <strong>the</strong> fact<br />

that <strong>the</strong> closed span is always complemented. Indeed, let Xn = ~iEFn aiei be a normalized<br />

block basis <strong>of</strong> (ei) (here <strong>the</strong> Fn are f<strong>in</strong>ite subsets <strong>of</strong> I~t and F1 < F2 < ... <strong>in</strong> <strong>the</strong> sense<br />

that maxFn < m<strong>in</strong> Fn+l)and def<strong>in</strong>e x n = IlXnll -p Y]icFn lai]P-2aie* for each n, where, as<br />

usual, e* denotes <strong>the</strong> biorthogonal functional to ei. Then (x*) is biorthogonal to (Xn). The<br />

formula<br />

OO<br />

Px -- E x* (x)xn forxeep<br />

n=l<br />

<strong>the</strong>n def<strong>in</strong>es a norm one projection from g p onto <strong>the</strong> closed span <strong>of</strong> (Xn), [71, Proposition<br />

2.a. 1 ], [44, Proposition 1.5.3]. This gives<br />

THEOREM 12. If (xn) is a block basis <strong>of</strong> g.p, <strong>the</strong>n <strong>the</strong>re is a norm one projection from g.p<br />

onto [Xn : n ~ H].<br />

As noted <strong>in</strong> Section 2, a very similar argument yields that if (Xi) ~ Lp is a nonzero<br />

disjo<strong>in</strong>tly supported normalized sequence <strong>the</strong>n (xi) is 1-equivalent to <strong>the</strong> unit vector basis<br />

<strong>of</strong> ep and [(xi)] is 1-complemented <strong>in</strong> Lp.<br />

The analog <strong>of</strong> Theorem 12 holds for co and, as it turns out, is peculiar to <strong>the</strong>se unconditional<br />

bases.<br />

THEOREM 13 ([69]). Suppose that X is a <strong>Banach</strong> space with an unconditional bases such<br />

that for every permutation rc <strong>of</strong> H and every block basis (yj) <strong>of</strong> (Xjr(n)), [yj: j ~ H] is<br />

complemented, <strong>the</strong>n (Xn) is equivalent to <strong>the</strong> unit vector basis <strong>of</strong> g.p or co.<br />

As was noted <strong>in</strong> [49, Section 4], it is an immediate consequence <strong>of</strong> Theorem 12 and<br />

standard perturbation arguments that<br />

COROLLARY 14. If Y is an <strong>in</strong>f<strong>in</strong>ite dimensional subspace <strong>of</strong> g.p, <strong>the</strong>n for every e > O, Y<br />

has a (1 + E)-complemented subspace (1 + e)-isomorphic to g.p.<br />

We turn to some results <strong>of</strong> Petczyfiski [86] which were important <strong>in</strong> stimulat<strong>in</strong>g <strong>in</strong>terest<br />

<strong>in</strong> classification problems for complemented subspaces.<br />

THEOREM 15. If X is an <strong>in</strong>f<strong>in</strong>ite dimensional complemented subspace <strong>of</strong> g.p, <strong>the</strong>n X is<br />

isomorphic to ( p.<br />

A pro<strong>of</strong> us<strong>in</strong>g Pe{czyfiski's decomposition method, [86] is given <strong>in</strong> [49, Section 4].<br />

A more effective but more complex pro<strong>of</strong> is given <strong>in</strong> [7]. That pro<strong>of</strong> proceeds by directly<br />

show<strong>in</strong>g that a complemented subspace <strong>of</strong> g p actually decomposes <strong>in</strong>to a sum <strong>of</strong> f<strong>in</strong>ite<br />

n<br />

dimensional subspaces each isomorphic to g p for an appropriate n and <strong>the</strong>n deduc<strong>in</strong>g that<br />

<strong>the</strong> space is isomorphic to g p from <strong>the</strong> decomposition.<br />

The pro<strong>of</strong> us<strong>in</strong>g <strong>the</strong> decomposition method depends ra<strong>the</strong>r heavily on <strong>the</strong> ability to estimate<br />

<strong>the</strong> norm <strong>of</strong> an operator on a g p sum by norms <strong>of</strong> <strong>the</strong> restrictions to <strong>the</strong> summands. In

136 D. Alspach and E. Odell<br />

Section 5 we will see that <strong>the</strong> fact that this does not work for o<strong>the</strong>r types <strong>of</strong> unconditional<br />

sums leads to a ra<strong>the</strong>r delicate approach to understand<strong>in</strong>g <strong>the</strong> complemented subspaces <strong>of</strong><br />

C<br />

Lp. It is not true <strong>in</strong> general that if X ~ Y and Y ~ X <strong>the</strong>n X is isomorphic to Y [42].<br />

One problem that was open until <strong>the</strong> 1970's was whe<strong>the</strong>r every <strong>Banach</strong> space conta<strong>in</strong>ed<br />

g p for some p or co. If we are satisfied with f<strong>in</strong>ite dimensional subspaces Dvoretzky's <strong>the</strong>orem<br />

[33] tells us that for every <strong>Banach</strong> space X, ~2 is f<strong>in</strong>itely represented <strong>in</strong> X. Kriv<strong>in</strong>e's<br />

<strong>the</strong>orem [61 ] extends this to basic sequences: Every basic sequence admits a p so that g p<br />

is block f<strong>in</strong>itely represented <strong>the</strong>re<strong>in</strong>. Go<strong>in</strong>g <strong>the</strong> o<strong>the</strong>r way if X is such that for some ~. < cx~,<br />

1 ~< p < ~, every f<strong>in</strong>ite dimensional subspace <strong>of</strong> X X-embeds <strong>in</strong>to ~p m for some m <strong>the</strong>n X<br />

embeds <strong>in</strong>to L p. This is easy via an ultraproduct argument. In any event if we are seek<strong>in</strong>g<br />

some collection <strong>of</strong> <strong>in</strong>f<strong>in</strong>ite dimensional "atoms" which must live <strong>in</strong>side <strong>of</strong> every X, <strong>the</strong>se<br />

must <strong>in</strong>clude more than just co and g p, 1 ~< p < cxz, as witnessed by Tsirelson's example<br />

[99,28]. However we do get a positive result for subspaces X <strong>of</strong> Lp (1 ~< p < c~). In fact<br />

with<strong>in</strong> a certa<strong>in</strong> class <strong>of</strong> "stable" spaces <strong>the</strong> conjecture is true <strong>in</strong> a strong sense. Before we<br />

discuss <strong>the</strong>se results let us def<strong>in</strong>e a notion that is a little stronger than f<strong>in</strong>ite representability.<br />

DEFINITION 16. Suppose that (Xn) is a sequence <strong>in</strong> a <strong>Banach</strong> space X and that Y is a<br />

<strong>Banach</strong> space with a basis (Yn). We say (Xn) generates <strong>the</strong> spread<strong>in</strong>g model (Y, (Yn)) if for<br />

every s > 0 and k E N, <strong>the</strong>re is an N such that<br />

(1 +/3) -1 ~aiYi<br />

k=l<br />

i--1<br />

aiXni ~ (1 + s) ai Yi<br />

k--1<br />

for all N < ni < n2 < ... < nk and f<strong>in</strong>ite sequences <strong>of</strong> scalars (ai). (Y, (Yn)) is said to be<br />

a spread<strong>in</strong>g model over X if <strong>the</strong>re is a sequence (xn) <strong>in</strong> X generat<strong>in</strong>g <strong>the</strong> spread<strong>in</strong>g model<br />

(Y, (y~)) and such that limnl

L p spaces 137<br />

DEFINITION 17. Suppose X is a separable <strong>Banach</strong> space. We say that X is stable if for<br />

any two bounded sequences (xn) and (yn) <strong>in</strong> X and free ultrafilters b/and 12 on N,<br />

lim lim Ilxn -+- Ym II - lim lim Ilxn -+- Ym II.<br />

(n)E/~ (m)EV<br />

(m) E'I2 (n) E/~<br />

It is easy to see that all f<strong>in</strong>ite dimensional spaces are stable. In <strong>the</strong> case <strong>of</strong> Hilbert space<br />

a simple computation with <strong>the</strong> <strong>in</strong>ner product establishes stability. Indeed, without loss <strong>of</strong><br />

generality let (xn) and (Yn) be weakly convergent sequences <strong>in</strong> a Hilbert space X with limits<br />

x and y, respectively. (We denote <strong>the</strong> <strong>in</strong>ner product by (., .).) Not<strong>in</strong>g that lim(n)cU Ilxn II<br />

and lim(m)~V [lYm I[ exist and<br />

lim lim IIx~ -+- Ym II 2 -- IIx~ II 2 - Ilym II 2 - 2 lim lim (xn, Ym)<br />

(n)c/g (m)E]2<br />

(n)ELr (m)EV<br />

= 2(x, y)- lim lim Ilxn + Ym II 2 -[IXnll 2 --IlYm 112,<br />

(m)EV (n)EU<br />

(3.1)<br />

we see that X is stable.<br />

Clearly any subspace <strong>of</strong> a stable space is stable. Maurey and Kriv<strong>in</strong>e proved that if X<br />

is a stable <strong>Banach</strong> space and 1

138 D. Alspach and E. Odell<br />

The convolution <strong>of</strong> two types r and or, realized by (Xn), bl and (Yn), 12 respectively, is<br />

def<strong>in</strong>ed by<br />

r * cr (x) = lim lim IIx + Xn + Ym II,<br />

(n) e/.d (m)cV<br />

where r and o- are realized by (Xn) and L/and (Ym) and V, respectively. If <strong>the</strong> space X<br />

is stable <strong>the</strong> convolution is commutative. Fur<strong>the</strong>rmore <strong>the</strong> def<strong>in</strong>ition <strong>the</strong>n does not depend<br />

on <strong>the</strong> particular realizations <strong>of</strong> r and o- and r 9 cr can be easily shown to itself be a type<br />

on X.<br />

A type r is said to be symmetric if r = -r, i.e., r (x) -- r(-x) for all x. Observe that<br />

if (Xn) is <strong>the</strong> fundamental sequence <strong>of</strong> a spread<strong>in</strong>g model which is isometric to g p over X<br />

<strong>the</strong>n <strong>the</strong> type realized by (Xn) and any ultrafilter/,4 is symmetric.<br />

DEFINITION 19. A symmetric type r is said to be an s if for any non-negative<br />

scalars a, b, ar * br = (a p + bp)I/Pr.<br />

Notice that if (Xn) is a weakly null sequence <strong>in</strong> s and limn---,~ Ilxn II = P, 1 < p < cxz,<br />

<strong>the</strong>n r(x) = limn~oc Ilxn + xll = (PP + IlxllP) 1/p, for all x e ep, is a symmetric type. Also<br />

(ar,br)(x) = lim lim Ilx + Xn +Xmll<br />

n--+ oc m--+ oo<br />

= (llxll p +aPp p -+-bPpP) 1/p- (a p nt-bp)l/Pr(x). (3.2)<br />

Thus r is an gp-type.<br />

Now suppose that (Xn) is a normalized sequence and L/is an ultrafilter that realize an<br />

p-type r on X. Then for any x e X,<br />

ql II<br />

lim lim ... lim x n t- aixni -- a l'r. a2"c -k . . . . akr(x)<br />

(nl)Eb/(n2) cb/ (nk)cld i--1<br />

- a r(x)- lim x+ a Xn 9<br />

(n)cld<br />

i=1<br />

(3.3)<br />

(3.4)<br />

Thus (Xn) is <strong>the</strong> generat<strong>in</strong>g sequence <strong>of</strong> a spread<strong>in</strong>g model over X which is isometric to<br />

(~p, (e,,)) over X.<br />

As noted above Kriv<strong>in</strong>e, [61], had earlier shown that <strong>in</strong> any basic sequence <strong>the</strong> space<br />

p, for some p, or co is block f<strong>in</strong>itely representable. Usually f<strong>in</strong>ite representability tells us<br />

little about <strong>the</strong> <strong>in</strong>f<strong>in</strong>ite dimensional structure <strong>of</strong> <strong>the</strong> space. Maurey and Kriv<strong>in</strong>e show that<br />

<strong>in</strong> a stable <strong>Banach</strong> space Kriv<strong>in</strong>e's <strong>the</strong>orem can be used to obta<strong>in</strong> much more.<br />

THEOREM 20. Suppose that X is a separable stable <strong>Banach</strong> space. Then for some 1

L p spaces 139<br />

THEOREM 21. Suppose that (Xn) is a sequence <strong>in</strong> a separable stable <strong>Banach</strong> space X<br />

which generates a spread<strong>in</strong>g model which is isometric to (s p, (en)) over X. Then for every<br />

e > 0 <strong>the</strong>re is a subsequence <strong>of</strong> (Xn) which is (1 + e)-equivalent to <strong>the</strong> basis <strong>of</strong> s<br />

The pro<strong>of</strong> <strong>of</strong> Theorem 21 is quite easy. Given e > 0 we choose suitable ek $ 0 and us<strong>in</strong>g<br />

a compactness argument <strong>in</strong>ductively choose a subsequence (x/7 i) <strong>of</strong> (x/7) satisfy<strong>in</strong>g" for all<br />

k and (/~i)'l __V_ [-1, 1],<br />

k-2<br />

~~j2~ + (l~k_l I p -+-I~klP)l/Pxn~_,<br />

j--I<br />

k-2<br />

-- ~ fljXnj + ([flk-I I p + ]flkl p) l/pel<br />

j=l<br />

< 6k<br />

and<br />

k<br />

Z<br />

j=l<br />

fljXnj<br />

k-2<br />

ZfljXnj + flk-lel + flke2<br />

j--I<br />

< 6k.<br />

Us<strong>in</strong>g <strong>the</strong>se two <strong>in</strong>equalities one can iteratively keep comb<strong>in</strong><strong>in</strong>g <strong>the</strong> last two terms <strong>in</strong><br />

y~] ~jxj to obta<strong>in</strong> <strong>the</strong> result.<br />

The pro<strong>of</strong> <strong>of</strong> Theorem 20 is more difficult. One first shows that a symmetric type cr on a<br />

stable space X is an s type for some 1 ~< p < oe iff for all u > 0 <strong>the</strong>re exists 15 ~> 0 with<br />

~r, (otcr) --/3cr. To get <strong>the</strong> existence <strong>of</strong> such acr is fairly <strong>in</strong>volved and employs <strong>the</strong> notion<br />

<strong>of</strong> conic classes <strong>of</strong> types.<br />

See <strong>the</strong> orig<strong>in</strong>al paper <strong>of</strong> Maurey and Kriv<strong>in</strong>e, [62,44,40] or [23] for pro<strong>of</strong>s. The last<br />

reference proves <strong>the</strong> existence <strong>of</strong> s by us<strong>in</strong>g an ord<strong>in</strong>al <strong>in</strong>dex approach which is<br />

quite different from any <strong>of</strong> <strong>the</strong> o<strong>the</strong>rs and is somewhat simpler. One uses Kriv<strong>in</strong>e's <strong>the</strong>orem<br />

<strong>in</strong> a certa<strong>in</strong> manner to obta<strong>in</strong> a p. Then given e > 0 one considers<br />

He,p(X) -- { (xi)~ ~ X: (xi)~ is (1 + e)-equivalent to <strong>the</strong> unit vector basis<br />

/7<br />

Of~p}.<br />

This is naturally a closed tree and can be given an ord<strong>in</strong>al <strong>in</strong>dex [ 16]. One shows that <strong>the</strong><br />

<strong>in</strong>dex is W l and so <strong>the</strong> tree has an <strong>in</strong>f<strong>in</strong>ite branch.<br />

In general, even for very well-behaved sequences <strong>in</strong> L p for 1 ~< p < 2, one cannot f<strong>in</strong>d<br />

a nice s basis by pass<strong>in</strong>g to subsequences. (See [43].) These results have been extended<br />

to weakly stable <strong>Banach</strong> spaces (def<strong>in</strong>ed as <strong>in</strong> Def<strong>in</strong>ition 17 except one also requires (Xn)<br />

and (Yn) to be weakly null) which allows for <strong>the</strong> case <strong>of</strong> co [12].<br />

Thus every subspace <strong>of</strong> a stable <strong>Banach</strong> space conta<strong>in</strong>s a subspace (1 + e)-isomorphic<br />

to s One way to look at <strong>the</strong>se results is to say that every subspace <strong>of</strong> L p, 1

140 D. Alspach and E. Odell<br />

<strong>in</strong>quire if <strong>the</strong>re are stronger results than Theorem 20. One such possibility is to replace X<br />

is stable by a condition implied by stability: <strong>the</strong> set <strong>of</strong> types on X, r (X), is separable <strong>in</strong> <strong>the</strong><br />

strong topology (that generated by <strong>the</strong> pseudometrics dM(a, r) = sup{la(x) -- r(x)l : ~ X,<br />

Ilxll ~< M}). Unfortunately <strong>the</strong> dual <strong>of</strong> Tsirelson's space [99] satisfies this condition so<br />

<strong>the</strong> attempt fails. However Haydon and Maurey [45] have shown that if r(X) is strongly<br />

separable <strong>the</strong>n X conta<strong>in</strong>s ei<strong>the</strong>r a reflexive subspace or an isomorph <strong>of</strong> ~1.<br />

One might also ask whe<strong>the</strong>r <strong>the</strong>re are o<strong>the</strong>r nice sequence spaces which embed isomorphically<br />

<strong>in</strong>to L p. The approach <strong>of</strong> types us<strong>in</strong>g convolution cannot be used <strong>in</strong> <strong>the</strong>se cases<br />

because <strong>the</strong>y depend on formulas which must give rise to an ~p basis, [15,72]. The situation<br />

for p > 2 is constra<strong>in</strong>ed by <strong>the</strong> gp - ~2 dichotomy (see Section 4 or [49, Section 4]),<br />

but for 1 ~< p < 2 <strong>the</strong>re is some room to maneuver. Dacunha-Castelle [30], showed that <strong>in</strong><br />

order for a symmetric sequence space to embed <strong>in</strong> L1, <strong>the</strong> space must be isomorphic to<br />

an average <strong>of</strong> Orlicz sequence spaces. Additional work was done by Kwapi6n and Schtitt<br />

[64,63], and Raynaud and Schtitt [88]. The pro<strong>of</strong>s <strong>of</strong> <strong>the</strong>se results are too technical for us<br />

to delve <strong>in</strong>to here, so we will content ourselves with stat<strong>in</strong>g a few <strong>of</strong> <strong>the</strong>m.<br />

THEOREM 22. There is a universal constant K such that if Y is a subspace <strong>of</strong> L1 with<br />

C-symmetric basis, <strong>the</strong>n <strong>the</strong>re is a probability space (I2, P) and a family <strong>of</strong> 2-concave<br />

normalized Orlicz functions F~o, o9 ~ I2 such that<br />

(KC)-If~ IlfllF~o dP ~< ]lfll ~< KCfs2 [ifl[Fo~dP<br />

for all f e Y.<br />

Bretagnolle and Dacunha-Castelle [22] showed that each Orlicz space with a 2-concave<br />

Orlicz function embeds <strong>in</strong>to L1. Thus <strong>the</strong>se two results give a pretty clear picture <strong>of</strong> <strong>the</strong><br />

symmetric sequence spaces which can embed <strong>in</strong> L1 and hence <strong>in</strong> Lp, for 1 < p < 2, [93].<br />

(See [53] for some additional material on symmetric and unconditional sequences <strong>in</strong> L1.)<br />

An example <strong>of</strong> a more general result from [88] is<br />

THEOREM 23. Lr<br />

is isomorphic to a subspace <strong>of</strong> L1 if and only if <strong>the</strong>re is a p such<br />

that Lr is order isomorphic to a sublattice <strong>of</strong> L1 (Lp) and X is isomorphic to a subspace<br />

<strong>of</strong> Lp.<br />

4. Subspaces <strong>of</strong> L p<br />

As noted <strong>in</strong> Section 3, if X c s <strong>the</strong>n for all e > 0, X conta<strong>in</strong>s Y with d (Y, s < 1 + e and<br />

Y is (1 + e)-complemented <strong>in</strong> s Not much more can be said about an arbitrary X c s<br />

X need not even have <strong>the</strong> approximation property, let alone a basis if p ~: 2, [26]. But we<br />

can say someth<strong>in</strong>g if X has an FDD.<br />

THEOREM 24 ([57]). Let (En) be an FDD for X c_ g~p. Then (En) can be blocked <strong>in</strong>to an<br />

g~ p decomposition (Fn). Thus X ~ (Y~ Fn ) p.

L p spaces 141<br />

We shall sketch <strong>the</strong> pro<strong>of</strong> which illustrates <strong>the</strong> block<strong>in</strong>g techniques developed by Johnson<br />

and Zipp<strong>in</strong> <strong>in</strong> several papers [57,56] and later fur<strong>the</strong>r developed [52]. This technique<br />

will have application for subspaces <strong>of</strong> L p. An extension <strong>of</strong> this <strong>the</strong>orem is given <strong>in</strong> Propo-<br />

sition 31. If A, B and C are positive numbers we write A c B if A ~< CB and B n 1 so that<br />

2-1-1<br />

I1(I - P~2)xll < 62<br />

if X E B(Ei)I2.<br />

Cont<strong>in</strong>ue <strong>in</strong> this manner to obta<strong>in</strong> subsequences (m i) and (n i) <strong>of</strong> 1~ satisfy<strong>in</strong>g <strong>the</strong> follow<strong>in</strong>g<br />

mn<br />

for Gn -- (Ei)i=mn_l+l<br />

(a) IIP~.xll < 6k if x E B[ci]i>~+:, k >~ 1, and<br />

(b) I1(I - P~)xll < ek ifx E B(Gz)~: ' , k~>l .<br />

A (possibly f<strong>in</strong>ite) sequence (x~) is a skipped block sequence with respect to (Gi) if<br />

<strong>the</strong>re exist <strong>in</strong>tegers r l

142 D. Alspach and E. Odell<br />

THEOREM 26. Let 1

Lp spaces 143<br />

The mapp<strong>in</strong>g T'[(Xn)] --+ (Y~ Lp(En))p given by Tx -- (x[En) is norm 1. The diagonal<br />

map D(y~ a~x~) -- (a~xn IEn) is bounded with ]]D[] ~< )~ [71, p. 20]. This yields <strong>the</strong> result.<br />

E3<br />

To obta<strong>in</strong> <strong>the</strong> "1 § e" conclusion <strong>of</strong> (c) we need to pass to a normalized block basis (yi)<br />

<strong>of</strong> (xi) which is a perturbation <strong>of</strong> a disjo<strong>in</strong>tly supported sequence. For example if p = 1 we<br />

may assume that xi = ui § di where (di) is disjo<strong>in</strong>tly supported and bounded away from 0<br />

<strong>in</strong> norm (hence equivalent to <strong>the</strong> unit vector basis <strong>of</strong> g l) and (u i) is uniformly <strong>in</strong>tegrable.<br />

This follows from <strong>the</strong> subsequence splitt<strong>in</strong>g lemma.<br />

THEOREM 29. If (fi) CC_ BLi <strong>the</strong>n <strong>the</strong>re exists a subsequence f/ -- ui § di where ]Ui] /~<br />

]di] -- 0 for all i, (u i) is uniformly <strong>in</strong>tegrable and (di) is a disjo<strong>in</strong>tly supported sequence.<br />

S<strong>in</strong>ce a uniformly <strong>in</strong>tegrable subset <strong>of</strong> L l is relatively weakly compact [49, Section 4],<br />

we may assume that (u i) is weakly convergent and hence cannot be equivalent to <strong>the</strong> unit<br />

vector basis <strong>of</strong> el. Thus some normalized block basis (yi) <strong>of</strong> (Xi) is a perturbation <strong>of</strong> a<br />

block basis <strong>of</strong> (di) which yields (c).<br />

D<br />

REMARK. While Hilbert subspaces <strong>of</strong> Lp (2 < p < cx~) are automatically complemented,<br />

for 1 < p < 2 every Hilbert subspace <strong>of</strong> L p conta<strong>in</strong>s a subspace which is complemented<br />

<strong>in</strong> Lp. Indeed let (xi) be a basis for X equivalent to <strong>the</strong> unit vector basis <strong>of</strong> ~2 with biorthog-<br />

1 1<br />

onal functionals (x*). Let Q'Lq --+ X* be <strong>the</strong> natural quotient map (p + q - 1). Let<br />

Qyi - x* with ([[yi [[) bounded. By Theorem 27(a) it follows that some subsequence <strong>of</strong><br />

(y~) must be equivalent to <strong>the</strong> unit vector basis <strong>of</strong> ~2 and [(y~)] is complemented <strong>in</strong> Lq by<br />

<strong>the</strong> orthogonal projection. It follows that [(x~)] is complemented <strong>in</strong> L p.<br />

A subspace <strong>of</strong> L p which is "small" embeds <strong>in</strong>to g p.<br />

THEOREM 30 ([51,46]). Let X c_ L p, 1 < p :fi 2 < cx~.<br />

(a) If2 < p < cxz and ~2 5/-+ X <strong>the</strong>n X ~ gp.<br />

(b) If 1 < p < 2 and <strong>the</strong>re exists K < ~ so that every normalized weakly null sequence<br />

<strong>in</strong> X has a subsequence K-equivalent to <strong>the</strong> unit vector basis <strong>of</strong> g.p <strong>the</strong>n X ~ g.p.<br />

Part (a) has been streng<strong>the</strong>ned [60] to yield that for all ~ > 0, X l_+~ s We shall sketch<br />

a unified approach to prov<strong>in</strong>g both parts. Let T~o -- {(nl ..... nk)" k E N and ni E • for<br />

1 ~< i ~< k} be <strong>the</strong> countably branch<strong>in</strong>g tree orderedby (nl ..... nk)

144 D. Alspach and E. Odell<br />

PROOF. By <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Theorem 24 (us<strong>in</strong>g Lemma 25) it suffices to show that (En) can<br />

be blocked <strong>in</strong>to a skipped gp decomposition, i.e., <strong>the</strong>re exists a block<strong>in</strong>g (Fn) <strong>of</strong> (En) and<br />

C < ec so that if (xn) is any normalized skipped block sequence <strong>of</strong> (Fn), <strong>the</strong>n (Xn) is C-<br />

equivalent to <strong>the</strong> unit vector basis <strong>of</strong> g p. C may be taken to be K + 1. By a compactness<br />

argument it suffices to prove <strong>the</strong> follow<strong>in</strong>g.<br />

(A) 3nl 'v'xl E S(Ei)~ 3n2 Vx2 G_ S(Ei)~ 3n3 Vx3 E S(Ei)~ ... so that (xi)~ is K-<br />

equivalent to <strong>the</strong> unit vector basis <strong>of</strong> gp.<br />

The formal negation <strong>of</strong> (A) is<br />

(B) Vnl 3Xl E S(Ei),, 1 Vn2 3x2 E S(Ei),~2 Vn3 3x3 E S(Ei),,~... so that (xi) is not K-<br />

equivalent to <strong>the</strong> unit vector basis <strong>of</strong> gp.<br />

S<strong>in</strong>ce <strong>the</strong>se are <strong>in</strong>f<strong>in</strong>ite sentences we cannot automatically assume that ei<strong>the</strong>r (A) or (B)<br />

must be true. However by Mart<strong>in</strong>'s <strong>the</strong>orem that Borel games are determ<strong>in</strong>ed [73] it does<br />

follow that (A) or (B) is true.<br />

Indeed consider a two player game where player (I) chooses n l E N and player (II)<br />

chooses X l E S(Ei)~. Then (I) chooses n2 > max(suppxl), <strong>the</strong> support be<strong>in</strong>g with respect<br />

to <strong>the</strong> Ei's. (II) chooses x2 E S/E i )n~ and so on. An outcome <strong>of</strong> this game is a sequence <strong>in</strong><br />

~-- {(nl,xl,n2, x2 .... )" Xj E S(Ei)nj and nj+l > max(suppxj) for j EN}.<br />

Player (I) w<strong>in</strong>s if <strong>the</strong> outcome belongs to<br />

r -- { (n 1, x l, n2, X2 .... ) E ~" (Xi)C~ is K-equivalent to <strong>the</strong> unit vector<br />

basis <strong>of</strong> g p }.<br />

The precise mean<strong>in</strong>g <strong>of</strong> (A) is that player (I) has a w<strong>in</strong>n<strong>in</strong>g strategy while (B) means that<br />

(II) has a w<strong>in</strong>n<strong>in</strong>g strategy. If ~ is given <strong>the</strong> relative product topology <strong>the</strong>n r is a closed<br />

subset <strong>of</strong> ~. Mart<strong>in</strong>'s <strong>the</strong>orem yields <strong>the</strong>n that this game is determ<strong>in</strong>ed, i.e., ei<strong>the</strong>r (I) or (II)<br />

has a w<strong>in</strong>n<strong>in</strong>g strategy.<br />

But if (B) holds one easily constructs a block tree T E To(X) with respect to (En) so<br />

that no branch is K-equivalent to <strong>the</strong> unit vector basis <strong>of</strong> ~ p.<br />

D<br />

By employ<strong>in</strong>g some fur<strong>the</strong>r block<strong>in</strong>g and perturbation arguments one can extend this<br />

result to<br />

PROPOSITION 32 ([84]). Let 1 < p < oc and let X be a reflexive space. Assume <strong>the</strong>re<br />

exists K < oo so that whenever T E T~o(X) is a weakly null tree <strong>the</strong>n some branch <strong>of</strong><br />

T is K-equivalent to <strong>the</strong> unit vector basis <strong>of</strong> g.p. Then <strong>the</strong>re exists a sequence <strong>of</strong> f<strong>in</strong>ite<br />

dimensional spaces (Fn) so that X ~ (~. Fn)ep.<br />

PROOF OF THEOREM 30. One first shows that <strong>the</strong> hypo<strong>the</strong>ses <strong>of</strong> both (a) and (b) imply<br />

<strong>the</strong> hypo<strong>the</strong>sis <strong>of</strong> Proposition 32. This depends upon <strong>the</strong> fact that X c L p and arguments<br />

like those used to prove Theorem 27.<br />

For example (b) yields that <strong>the</strong>re exists 3 -- 3(K) > 0 so that if (xi) is a normalized<br />

f<br />

weakly null sequence <strong>in</strong> X <strong>the</strong>n some subsequence can be written as x i -- Yi -Jr- di where

L p spaces 145<br />

Yi /X di -- 0, <strong>the</strong> di's are disjo<strong>in</strong>tly supported and Ildi I1 ~> 6 for all i while (a) yields <strong>in</strong><br />

addition that one can take []di ]1 --+ 1. Us<strong>in</strong>g this given T ~ T~o(X) one can select a branch<br />

<strong>of</strong> <strong>the</strong> tree exhibit<strong>in</strong>g similar behavior and hence is C (K)-equivalent to <strong>the</strong> unit vector basis<br />

Of~p.<br />

The pro<strong>of</strong> <strong>of</strong> Proposition 32 yields that if X c Y, where Y is reflexive with an FDD (En)<br />

<strong>the</strong>n one can block (En) <strong>in</strong>to (Fn) so that X ~ (~ Fn)ep. Thus one can block <strong>the</strong> Haar<br />

basis <strong>in</strong>to an FDD (Fn) so that X embeds <strong>in</strong>to (~ Fn)ep. S<strong>in</strong>ce each Fn ~ gp" for some<br />

m n ~ N, <strong>the</strong> latter space is a subspace <strong>of</strong> ~p.<br />

ff]<br />

From Theorems 30 and 27 it follows that a subspace X <strong>of</strong> L p for 2 < p < oc which<br />

is not isomorphic to ~2 nor to a subspace <strong>of</strong> s must conta<strong>in</strong> ~2 | g p. What additional<br />

conditions on X ensure that X ~ g p ff~ g2 ? The follow<strong>in</strong>g gives one such result.<br />

THEOREM 33 ([52]). Let 2 < p < oc. Let X be a subspace <strong>of</strong> Lp which is isomorphic to<br />

a quotient <strong>of</strong>a subspace <strong>of</strong>g~p • ~2. Then X embeds <strong>in</strong>to g~p 9 g~2.<br />

The method <strong>of</strong> pro<strong>of</strong> is a more complicated block<strong>in</strong>g argument which ultimately produces<br />

a block<strong>in</strong>g (Hn) <strong>of</strong> <strong>the</strong> Haar basis so that <strong>the</strong> natural mapp<strong>in</strong>g<br />

P<br />

is an <strong>in</strong>to isomorphism.<br />

The follow<strong>in</strong>g is open. It concerns <strong>the</strong> next layer <strong>of</strong> small subspaces <strong>of</strong> Lp.<br />

PROBLEM 34. Let X be a subspace <strong>of</strong> Lp, 2 < p < oc and assume that (~s ~ X.<br />

Does X ~ s 9 s<br />

The situation for subspaces <strong>of</strong> L p with 1 < p < 2 is <strong>of</strong> course more complicated (cf.<br />

Theorem 24). But we can say <strong>the</strong> follow<strong>in</strong>g.<br />

THEOREM 35 ([93]). Let 1

146 D. Alspach and E. Odell<br />

where d v = ~b dm. Thus s<strong>in</strong>ce x --+ x is an isometry <strong>of</strong> L1 with L l(v), <strong>the</strong> <strong>the</strong>orem will<br />

follow provided we have<br />

Step 2: Iq (X) < cx~ for some 1 < q ~< 2.<br />

If not one shows that s is f<strong>in</strong>itely representable <strong>in</strong> X. This yields that Bx is not uniformly<br />

<strong>in</strong>tegrable and hence X is not reflexive.<br />

D<br />

5. s 1 < p < cx~, p 7~ 2<br />

Recall that X is a s if and only if <strong>the</strong>re is a constant )~ < ~ such that for each<br />

f<strong>in</strong>ite dimensional subspace Z <strong>of</strong> X <strong>the</strong>re is a f<strong>in</strong>ite dimensional subspace Y <strong>of</strong> X such<br />

/7<br />

that d(Y, s p)

L p spaces 147<br />

realize this space as (y-~n~ 1 Lp (2-n, 2 -n+l) p. Thus by <strong>the</strong> decomposition method any<br />

complemented subspace <strong>of</strong> L p which conta<strong>in</strong>s L p complemented is isomorphic to Lp. In<br />

particular L p G g.p(F), where F is a f<strong>in</strong>ite or countably <strong>in</strong>f<strong>in</strong>ite set, is isomorphic to L p.<br />

It follows that <strong>the</strong>se two spaces are all <strong>of</strong> <strong>the</strong> isomorphic types that we get as <strong>the</strong> range <strong>of</strong> a<br />

conditional expectation onto a (non-f<strong>in</strong>ite) sub-cr-algebra <strong>of</strong> <strong>the</strong> Lebesgue sets. A slightly<br />

more general class <strong>of</strong> projections are those <strong>of</strong> <strong>the</strong> form Pf -- gE(fg -1 lsuppg I~) where<br />

g E L p and/3 is a sub-a-algebra. This is <strong>the</strong> general form <strong>of</strong> a contractive projection, [10,<br />

100]. These do not give us any new spaces s<strong>in</strong>ce <strong>the</strong> range <strong>of</strong> such a projection is isometric<br />

to L p (#) for some measure #. We have already seen that <strong>the</strong> sequence <strong>of</strong> Rademachers or<br />

Gaussians span complemented subspaces isomorphic to g2. It is easy to see <strong>the</strong>n that <strong>the</strong>re<br />

are complemented subspaces <strong>of</strong> Lp which are isomorphic to ~p @ ~2 and (y~ ~2)p.<br />

A natural question is whe<strong>the</strong>r L p can be split <strong>in</strong>to two spaces which are not isomorphic<br />

to L p, i.e., are <strong>the</strong>re complemented subspaces X and Y <strong>of</strong> L p such that L p -- X 9 Y<br />

but nei<strong>the</strong>r X nor Y is isomorphic to L p? As we have seen <strong>in</strong> <strong>the</strong> case <strong>of</strong> L p (and s<br />

(Theorem 7 and Theorem 15) <strong>the</strong> answer is no. The fact that L p is primary tell us "half"<br />

<strong>of</strong> <strong>the</strong> s but little about <strong>the</strong> far more <strong>in</strong>terest<strong>in</strong>g o<strong>the</strong>r "half". Let us now consider<br />

<strong>the</strong> one complemented subspace <strong>of</strong> L p which is not a s<br />

Suppose that X is a subspace <strong>of</strong> Lp which is isomorphic to ~2. By [58] if p > 2, <strong>the</strong> Lp<br />

and L2 norms are equivalent and <strong>the</strong> orthogonal projection is bounded. If p < 2 it is no<br />

longer <strong>the</strong> case that a subspace isomorphic to ~2 is complemented. This was first proved<br />

for certa<strong>in</strong> values <strong>of</strong> p by us<strong>in</strong>g some results from harmonic analysis, [13, p. 52], [94,20].<br />

Later a completely different approach yielded <strong>the</strong> existence <strong>of</strong> uncomplemented subspaces<br />

<strong>of</strong> Lp, for each p, 1 < p < 2, which are isomorphic to g2, [13]. This immediately implies<br />

<strong>the</strong> existence <strong>of</strong> subspaces <strong>of</strong> L p which are isomorphic to e p and not complemented, s<strong>in</strong>ce<br />

g p is isomorphic to (~ g~)p. For 2 < p < oc, Rosenthal showed that <strong>the</strong>re are uncomplemented<br />

subspaces <strong>of</strong> g p which are isomorphic to g p, [91, Theorem 6]. Curiously, such a<br />

subspace cannot be <strong>the</strong> span <strong>of</strong> a sequence <strong>of</strong> <strong>in</strong>dependent random variables, [31,17]. For<br />

some time it was thought that <strong>the</strong> situation might be different for p - 1, but <strong>in</strong> [ 17] Bourga<strong>in</strong><br />

shows that <strong>the</strong>re are subspaces <strong>of</strong> L1 isomorphic to el which are uncomplemented.<br />

If we look at projections <strong>the</strong>re is one very special property <strong>of</strong> g2 subspaces <strong>of</strong> L p,<br />

1 < p < 2. While it may not be true that <strong>the</strong> subspace itself is complemented, as we have<br />

seen, for any subspace X <strong>of</strong> L p which is isomorphic to g z, <strong>the</strong>re is an <strong>in</strong>f<strong>in</strong>ite dimensional<br />

subspace Y <strong>of</strong> X which is complemented. If X is a subspace <strong>of</strong> L p which is isomorphic to<br />

p, <strong>the</strong>n as was shown <strong>in</strong> Section 3 it must conta<strong>in</strong> a subspace which is complemented (and<br />

necessarily isomorphic to gp). It follows that for a subspace <strong>of</strong> Lp which is isomorphic to<br />

~p (~ ~2 <strong>the</strong>re is a subspace isomorphic to ~p • ~2 which is complemented <strong>in</strong> Lp. This<br />

sort <strong>of</strong> hereditary complementation property does not hold for general complemented subspaces<br />

<strong>of</strong> L p. Consequently, a special class <strong>of</strong> operators has been <strong>in</strong>troduced which have<br />

somewhat better properties with respect to g2 subspaces. This concept occurred implicitly<br />

<strong>in</strong> [91] and <strong>the</strong>n was developed fur<strong>the</strong>r <strong>in</strong> several papers, [7,5,6,38,52]. In [58] it was<br />

shown that subspaces <strong>of</strong> L p, 2 < p < oc, which are isomorphic to/~2 are complemented<br />

by show<strong>in</strong>g that <strong>the</strong> orthogonal projection acts as a bounded operator on L p. This is an<br />

example <strong>of</strong> an operator which is simultaneously bounded <strong>in</strong> both norms on L p. This, as it<br />

turns out, is not really an unusual situation.

148 D. Alspach and E. Odell<br />

THEOREM 37 ([47,53]). Suppose that 1

L p spaces 149<br />

In order to deal with spaces with a norm def<strong>in</strong>ed <strong>in</strong> terms <strong>of</strong> an L2 norm and an Lp<br />

norm and with similar subspaces <strong>of</strong> Lp, we will use <strong>the</strong> terms (p, 2)-bounded, (p, 2)-<br />

isomorphism, etc., to <strong>in</strong>dicate that <strong>the</strong> map is bounded, is an isomorphism, etc., <strong>in</strong> both<br />

norms. For example, if T is a map from (~ Xn)(p,2,(w,,)) <strong>in</strong>to Lp, we would say that<br />

T is (p, 2)-bounded with IlTllp,2 ~< K, if <strong>the</strong>re exists a constant K such that IlT(xn)ll2 ~<<br />

2 1/2<br />

K(y~ Ilxnll~w~)<br />

and IIT(x~)llp

150 D. Alspach and E. Odell<br />

The complemented subspaces <strong>of</strong> (Y]~ ~2)p have also been classified, [83]. It turns out that<br />

only <strong>the</strong> obvious spaces occur, g2, gp, gp (9 ~2 and (}-~' g2)p. The pro<strong>of</strong> is ra<strong>the</strong>r technical<br />

and ma<strong>in</strong>ly consists <strong>of</strong> carefully dist<strong>in</strong>guish<strong>in</strong>g between complemented subspaces which<br />

are isomorphic to complemented subspaces <strong>of</strong> gp | ~2 and those that conta<strong>in</strong> (y]~ g2)n-<br />

In 1972 Rosenthal proved his <strong>in</strong>equality and used it to show that <strong>the</strong>re were several o<strong>the</strong>r<br />

s<br />

In particular he def<strong>in</strong>ed <strong>the</strong> space X p which had some surpris<strong>in</strong>g properties<br />

and led to <strong>the</strong> construction [21] <strong>of</strong> uncountably many isomorphically dist<strong>in</strong>ct s<br />

Because X p has proved to be a fundamental space we will present some <strong>of</strong> Rosenthal's<br />

results <strong>in</strong> detail.<br />

To construct <strong>the</strong> space X p for p > 2, we let Xn be <strong>the</strong> one-dimensional space spanned<br />

by 1[0,1] for all n ~ N, and let (wn) be a sequence <strong>of</strong> numbers <strong>in</strong> (0, 1] such that for<br />

every e > 0, ~n:wn 0<br />

<strong>the</strong>n (~ Ixn [ 2 Wn) 2 1/2 >~ (<strong>in</strong>fwn)(y]~[xnlP) lip so that <strong>the</strong> norm is equivalent to <strong>the</strong> g2 norm.<br />

There is also a comb<strong>in</strong>ed case where <strong>the</strong> sequence breaks <strong>in</strong>to two pieces and thus yields<br />

~p (~ ~2.<br />

We have two issues to resolve with X p at this po<strong>in</strong>t. How do we see that it is a s<br />

space and how do we see that we really have only one isomorphism class? The solution to<br />

both <strong>in</strong>volves <strong>the</strong> construction <strong>of</strong> (p, 2)-bounded projections. (Also note that we can <strong>the</strong>n<br />

def<strong>in</strong>e Xq for q < 2 to be <strong>the</strong> dual <strong>of</strong> X p where p and q are conjugate <strong>in</strong>dices.) We will<br />

, ( OO<br />

use <strong>the</strong> representation above for (y]' Xn)(p,2 (w,,)) <strong>in</strong> Lp 1-1,,=1 [0, 1]), where <strong>the</strong> X,, are<br />

one-dimensional spaces spanned by a function Xn whose modulus is an <strong>in</strong>dicator function.<br />

For each n E N let x n = ]lxn 1122Xn and notice that Px - y]~n~ (f x n X)Xn is <strong>the</strong> orthogonal<br />

projection onto <strong>the</strong> closed span <strong>of</strong> (Xn) <strong>in</strong> L2. Thus P is bounded <strong>in</strong> <strong>the</strong> Lz-norm. Moreover<br />

by Rosenthal's <strong>in</strong>equality (1.11)<br />

n=l<br />

~< C max {<br />

n=l<br />

1"<br />

, Ix~,xl IIx. II p 9<br />

n=l<br />

Because P is bounded <strong>in</strong> <strong>the</strong> L2-norm and p > 2, <strong>the</strong> first expression <strong>in</strong> <strong>the</strong> max is bounded<br />

by Ilx l] p. To see that <strong>the</strong> second is also notice that<br />

n=l<br />

x*~x IIx. llp

L p spaces 151<br />

Because Ixn I is an <strong>in</strong>dicator function, Ilxn II 2 - IIx~ II p and thus (Xn Ilxn 1122 Ilxn II p) is a norm<br />

one sequence <strong>of</strong> mean zero symmetric <strong>in</strong>dependent random variables <strong>in</strong> Lq, where q --<br />

p/(p- 1). Consequently, <strong>the</strong> sequence is unconditional and has an upper s estimate.<br />

Therefore<br />

n--I<br />

f x, x IIx, 112 2 IIx, II p<br />

oo<br />

- fx~b~x~llx~l12211x~llp<br />

n-- 1<br />

~< IIx IlpC Ib, I q<br />

n--I<br />

for some constant C. This proves that <strong>the</strong> second expression is also bounded.<br />

The argument just given has been generalized to replace <strong>the</strong> one-dimensional spaces by<br />

<strong>the</strong> range <strong>of</strong> an orthogonal projection which is bounded on Lp. See [38, Theorem 4.8].<br />

THEOREM 39. Let 2 < p < cx~ and let (wn) be a sequence <strong>of</strong> scalars <strong>in</strong> (0, 1]. For each<br />

n let Xn be a subspace <strong>of</strong> mean zero functions <strong>in</strong> L p such that <strong>the</strong> orthogonal projection<br />

from Lp onto X,, Pn, is boundedand supllPnll < cx~. Then (~-~ Xn)(p,2,(w,,)) is isomorphic<br />

to a complemented subspace <strong>of</strong> L p.<br />

The pro<strong>of</strong> <strong>of</strong> this generalization is very similar to <strong>the</strong> pro<strong>of</strong> given above. The ma<strong>in</strong> po<strong>in</strong>t<br />

is to realize that <strong>the</strong> pairs (x,, x*) can be replaced by orthogonal projections and <strong>the</strong>ir<br />

adjo<strong>in</strong>ts.<br />

Next we will use a version <strong>of</strong> <strong>the</strong> decomposition method to show that <strong>the</strong>re is really<br />

only one isomorphism class for <strong>the</strong> spaces X(p,Z,(w,,) ) with (w,) thick near 0. To use <strong>the</strong><br />

decomposition method we need a suitable <strong>in</strong>f<strong>in</strong>ite sum <strong>of</strong> spaces. Rosenthal, [91 ], called<br />

this a symmetric sum; <strong>in</strong> our term<strong>in</strong>ology this is a (p, 2, (1)) sum, i.e., <strong>the</strong> sequence (wn)<br />

is just <strong>the</strong> constantly 1 sequence. As we noted earlier a key feature <strong>of</strong> <strong>the</strong> s that<br />

is used <strong>in</strong> <strong>the</strong> decomposition method is that one can comb<strong>in</strong>e a sequence <strong>of</strong> uniformly<br />

bounded operators on <strong>the</strong> <strong>in</strong>dividual summands to create an operator on <strong>the</strong> sum. For <strong>the</strong><br />

(p, 2, (1))-sum we have a similar property.<br />

LEMMA 40 ([6, Lemma 2.51). If {Xn} and {Yn} are two sequences <strong>of</strong> subspaces <strong>of</strong><br />

Lp(S2, #) for some probability measure #, and <strong>the</strong>re is a constant K and (p, 2)-<br />

cont<strong>in</strong>uous operators {Tn} such that I]Tnllp,2

- y*(Eanen)y<br />

152 D. Alspach and E. Odell<br />

affected by redundancy. Thus we can take a thick sequence (Wn) and def<strong>in</strong>e Wm,n = Wn<br />

for all n, m <strong>in</strong> N, to get a space X = X(p,Z,(wm,n)) which is obviously (p, 2)-isomorphic<br />

to <strong>the</strong> (p, 2, (1)) sum <strong>of</strong> <strong>in</strong>f<strong>in</strong>itely many copies <strong>of</strong> itself. Now we show how to construct<br />

projections which are (p, 2)-bounded.<br />

As Rosenthal discovered one can take a particular comb<strong>in</strong>ation <strong>of</strong> any f<strong>in</strong>ite set <strong>of</strong><br />

weights (Wn)ncF and get a block <strong>of</strong> <strong>the</strong> natural basis (en) <strong>of</strong> X(p,2,(Wn)) and a correspond-<br />

<strong>in</strong>g functional which is (p, 2)-cont<strong>in</strong>uous. If Zn6F It~2p/(p-2) ~ 1, let<br />

( )-'<br />

Y -- E'wn2/(p-2)D~n and y* - E w2P/(P-2) Ell) n2(p-1)/(p-2)e,n.<br />

Now notice that <strong>the</strong> map<br />

ncF n~F nEF<br />

Pn(Eanen) -<br />

E w2(p-1)/(P-2)an) y<br />

n~F<br />

is (p, 2)-cont<strong>in</strong>uous. Indeed,<br />

(~<br />

w2P/(P -2))'(z ll02(p-1)/(P-Z)an ) y<br />

n6F<br />

( z ) '1~ 2~ "~ 2'a~ (Z - 2~-2'<br />

_ ll)2p/(P -2) Wn 113n<br />

ncF n~F n6F<br />

,j2<br />

L p spaces 153<br />

for all k is (p, 2)-equivalent to <strong>the</strong> natural basis <strong>of</strong> X(p,2,(vn)) where<br />

v.-( ~,an,2<br />

w.<br />

2)'/2/(<br />

~_~ la.I p<br />

),/p<br />

n~Fk<br />

n6Fk<br />

2/(p-2)<br />

Notice that for <strong>the</strong> special case an -- ton<br />

)(p-2)/(2p)<br />

Z to2p/(p-2)<br />

n~Fk<br />

<strong>the</strong> weight vk is<br />

Also if K1 and K2 are positive constants and (Wn) and (W~n) are two sequences <strong>in</strong> (0, 1]<br />

such that KlWn

154 D. Alspach and E. Odell<br />

and is uniformly p-<strong>in</strong>tegrable. Us<strong>in</strong>g this Schechtman showed that @kn= 1 X p conta<strong>in</strong>s <strong>the</strong><br />

k-fold iterated sum<br />

(Z(Z (Z'r')r2<br />

where 2 ~> rl > r2 > ... > rk ~> p, but not a (k + 1)-fold iterated sum. Thus <strong>the</strong> spaces<br />

1 Xp, k - , 1 2,.. ., are different.<br />

The tensor product has one serious drawback <strong>in</strong> that <strong>the</strong> norm <strong>of</strong> <strong>the</strong> tensor product <strong>of</strong><br />

<strong>the</strong> projections grows rapidly. (See [6, Chapter 1] for some details.) Thus <strong>the</strong>re was no<br />

obvious way to use this to produce uncountably many different separable/2p-spaces. In<br />

[21,19] a completely different approach us<strong>in</strong>g <strong>in</strong>dependent sums was used to produce COl<br />

dist<strong>in</strong>ct/2p-spaces.<br />

In order to construct/2p-spaces based on <strong>in</strong>dependent sums without hav<strong>in</strong>g difficulties<br />

with constants, Bourga<strong>in</strong>, Rosenthal and Schechtman <strong>in</strong>troduce an isomorph <strong>of</strong> L p which<br />

is convenient for <strong>the</strong> construction. Let T denote <strong>the</strong> usual dyadic tree and for each node N<br />

<strong>of</strong> 7- we consider a copy <strong>of</strong> {0, 1 }, X2N, with <strong>the</strong> measure <strong>of</strong> each po<strong>in</strong>t one half. Now we<br />

def<strong>in</strong>e a certa<strong>in</strong> subspace L <strong>of</strong> L p on <strong>the</strong> natural product algebra and probability measure<br />

on I-IN~'T ~(2N, which is generated by those functions which are measurable with respect<br />

to any branch sub-a-algebra. That is, if B is a sequence <strong>of</strong> pairwise comparable nodes<br />

<strong>in</strong> <strong>the</strong> tree, <strong>the</strong>n we consider <strong>the</strong> sub-o'-algebra generated by sets which depend only on<br />

<strong>the</strong> coord<strong>in</strong>ates <strong>in</strong> B and L conta<strong>in</strong>s <strong>the</strong> functions which are measurable with respect to<br />

each such sub-o'-algebras. It is obvious that this space conta<strong>in</strong>s complemented subspaces<br />

isomorphic to L p <strong>of</strong> <strong>the</strong> Cantor set (with <strong>the</strong> usual product measure) because <strong>the</strong> o'-algebra<br />

generated by one branch is isomorphic to <strong>the</strong> Cantor a-algebra. Less obvious is <strong>the</strong> fact<br />

that this subspace L is complemented and thus isomorphic to L p.<br />

By consider<strong>in</strong>g subspaces <strong>of</strong> L, Rp, which are generated by functions which depend on<br />

certa<strong>in</strong> subtrees <strong>of</strong> 7- <strong>of</strong> order c~, Bourga<strong>in</strong>, Rosenthal and Schechtman show that <strong>the</strong>re are<br />

uncountably many s<br />

Their argument is actually to some extent an existence argument.<br />

The pro<strong>of</strong> depends on an ord<strong>in</strong>al Lp-<strong>in</strong>dex which <strong>the</strong>y cannot compute precisely<br />

but are able to show grows with <strong>the</strong> ord<strong>in</strong>al <strong>in</strong>dex <strong>of</strong> <strong>the</strong> subtree. In [6] <strong>the</strong> precise isomorphic<br />

classification <strong>of</strong> <strong>the</strong>se spaces is given. It is also shown <strong>the</strong>re that <strong>the</strong>se spaces are not<br />

isomorphic to any <strong>of</strong> those constructed by Schechtman. More details about <strong>the</strong> ord<strong>in</strong>al L p<br />

<strong>in</strong>dex can be found <strong>in</strong> [11 ].<br />

For <strong>the</strong> case p = 1, Bourga<strong>in</strong> [ 18,19] adapted some <strong>of</strong> <strong>the</strong> ideas from [21 ] to produce an<br />

uncountable family <strong>of</strong> s with <strong>the</strong> Radon-Nikodym property. The difficulty was<br />

that <strong>the</strong> analogous spaces for p = 1 are not complemented so Bourga<strong>in</strong> showed that <strong>the</strong>re<br />

are envelop<strong>in</strong>g/21-spaces which are complemented. Earlier Johnson and L<strong>in</strong>denstrauss,<br />

[48], showed by a completely different approach that <strong>the</strong>re are uncountably (actually 2 s~<br />

many separable non-isomorphic s which have <strong>the</strong> Radon-Nikodym and Schur<br />

properties but do not embed <strong>in</strong> separable dual spaces.<br />

At this po<strong>in</strong>t a complete isomorphic classification <strong>of</strong> <strong>the</strong> separable s seems out<br />

<strong>of</strong> reach. Most <strong>of</strong> <strong>the</strong> known relations among <strong>the</strong> smaller spaces can be found <strong>in</strong> [38]. The<br />

space X p itself has resisted study, [5]. It is not known whe<strong>the</strong>r <strong>the</strong>re are any complemented<br />

subspaces <strong>of</strong> Xp o<strong>the</strong>r than gp, ~2, ~p 9 ~2, and Xp or any o<strong>the</strong>r s<br />

except<br />

under additional assumptions such as hav<strong>in</strong>g an unconditional basis, [52].

L p spaces 155<br />

As we have noted above <strong>the</strong> general s<br />

are mysterious but <strong>the</strong> contractively<br />

complemented ones are just Lp(l z) spaces. This raises <strong>the</strong> question as to what is <strong>the</strong> size <strong>of</strong><br />

<strong>the</strong> largest constant 9/such that if P is a projection <strong>of</strong> norm less than y, <strong>the</strong>n <strong>the</strong> range <strong>of</strong><br />

P is isomorphic to L p (#) for some #. Because <strong>the</strong> norms <strong>of</strong> projections are l<strong>in</strong>ked to <strong>the</strong><br />

parameter )~ <strong>in</strong> <strong>the</strong> orig<strong>in</strong>al def<strong>in</strong>ition <strong>of</strong> <strong>the</strong> s<br />

<strong>the</strong>re is a companion question as<br />

to <strong>the</strong> largest constant )~ so that if X is a s<br />

<strong>the</strong>n X is isomorphic to Lp(#). One<br />

can also consider <strong>the</strong> question <strong>of</strong> <strong>the</strong> largest constant )~ such that if X is a s<br />

<strong>of</strong> L p <strong>the</strong>n X is complemented <strong>in</strong> Lp.<br />

In [32] it was shown that <strong>the</strong>re is a constant )~p such that if X is a f<strong>in</strong>ite dimensional<br />

/7<br />

subspace <strong>of</strong> L p and T is an isomorphism from s with standard basis (e j) onto X with<br />

][TI[II T-1 l[- )~ < )~p <strong>the</strong>n <strong>the</strong>re are disjo<strong>in</strong>t sets and e > 0 such that IlTejla j II > e for<br />

each j. Moreover e converges to 1 as )~ converges to 1. For <strong>the</strong> case p = 1 this was sufficient<br />

to imply that X is complemented. The ma<strong>in</strong> lemma <strong>in</strong> Dor's pro<strong>of</strong> is <strong>the</strong> follow<strong>in</strong>g.<br />

LEMMA 43. Let gl,g2 .... be a sequence <strong>of</strong> non-negative functions <strong>in</strong> L1 and let<br />

O

156 D. Alspach and E. Odell<br />

In <strong>the</strong> case p = 1 it is sufficient to have a uniform bound on <strong>the</strong> <strong>in</strong>dividual errors<br />

[Ifnl[O, 1]\ai II <strong>in</strong> order to show that <strong>the</strong> obvious candidate for a projection is bounded. For<br />

p > 1 Schechtman [97] was able to obta<strong>in</strong> an error estimate <strong>of</strong> <strong>the</strong> form<br />

II~ajTejlAj II a()~, e)(~ lajlP) '/p<br />

with a0~, e) --+ 0 as )~ ~ 1 and ~ --+ 1 which gave <strong>the</strong> complementation result <strong>in</strong> this<br />

case also. Thus if X is a/2p,Z subspace <strong>of</strong> L p, 1 < p < c~, with )~ sufficiently close to<br />

1, by pass<strong>in</strong>g to a limit <strong>of</strong> projections on an <strong>in</strong>creas<strong>in</strong>g family <strong>of</strong> subspaces Xn we get<br />

that X is complemented. (See, for example, [67].) For <strong>the</strong> case p = 1 pass<strong>in</strong>g to a limit is<br />

more problematical because <strong>the</strong> spaces are not reflexive. None<strong>the</strong>less it was shown <strong>in</strong> [8]<br />

that a/~l,l+e subspace <strong>of</strong> L1 is complemented if ~ is sufficiently small. Recently [41] a<br />

somewhat different approach to <strong>the</strong>se problems for p = 1 has been given which makes use<br />

<strong>of</strong> a representation <strong>the</strong>orem <strong>of</strong> Kalton for operators on Lp, 0 < p

Lp spaces 157<br />

[11]<br />

[121<br />

[]3]<br />

[141<br />

[151<br />

[161<br />

[171<br />

[18]<br />

[19]<br />

[20]<br />

[211<br />

[22]<br />

[231<br />

[241<br />

[251<br />

[26]<br />

[27]<br />

[281<br />

[291<br />

[301<br />

[3]]<br />

[32]<br />

[33]<br />

[341<br />

[351<br />

[36]<br />

[371<br />

[38]<br />

[391<br />

[40]<br />

[41]<br />

[42]<br />

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Israel J. Math. 8 (1970), 273-303.<br />

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CHAPTER 4<br />

Convex <strong>Geometry</strong> and Functional Analysis<br />

Keith Ball<br />

Department <strong>of</strong> Ma<strong>the</strong>matics, University College London, London, England<br />

E-mail: kmb @ math. ucl. ac. uk<br />

Contents<br />

Introduction ..................................................... 163<br />

1. Convolution <strong>in</strong>equalities <strong>in</strong> convex geometry ................................. 163<br />

1.1. The Brascamp-Lieb <strong>in</strong>equality ...................................... 164<br />

1.2. The reverse isoperimetric <strong>in</strong>equality ................................... 169<br />

1.3. The reverse Brascamp-Lieb <strong>in</strong>equality ................................. 171<br />

1.4. Higher dimensional projections ..................................... 173<br />

2. Volumes <strong>of</strong> sections <strong>of</strong> convex bodies ..................................... 174<br />

2.1. Vaaler's Theorem and its relatives .................................... 177<br />

2.2. Upper bounds for <strong>the</strong> volumes <strong>of</strong> sections <strong>of</strong> cubes ........................... 181<br />

3. Plank problems ................................................. 182<br />

3.1. Bang's lemma ............................................... 185<br />

3.2. The aff<strong>in</strong>e plank <strong>the</strong>orem ......................................... 185<br />

3.3. Nazarov's solution to <strong>the</strong> coefficient problem .............................. 189<br />

3.4. Related results and open problems .................................... 192<br />

References ..................................................... 193<br />

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 1<br />

Edited by William B. Johnson and Joram L<strong>in</strong>denstrauss<br />

9 2001 Elsevier Science B.V. All rights reserved<br />

161

Convex geometry and functional analysis 163<br />

Introduction<br />

This article describes three topics that lie at <strong>the</strong> <strong>in</strong>tersection <strong>of</strong> functional analysis, harmonic<br /&