Basic Concepts in the Geometry of Banach Spaces

Preface
1. Introduction
The aim of this Handbook is to present an overview of the main research directions and results
in Banach space theory obtained during the last half century. The scope of the theory,
having widened considerably over the years, now has deep and close ties with many areas
of mathematics, including harmonic analysis, complex analysis, partial differential equations,
classical convexity, probability theory, combinatorics, logic, approximation theory,
geometric measure theory, operator theory, and others. In choosing a topic for an article in
the Handbook we considered both the interest the topic would have for non-specialists as
well as the importance of the topic for the core of Banach space theory, which is the study
of the geometry of infinite dimensional Banach spaces and n-dimensional normed spaces
with n finite but large (local theory).
Many of the leading experts on the various aspects of Banach space theory have written
an exposition of the main results, problems, and methods in areas of their expertise. The
enthusiastic response we received from the community was gratifying, and we are deeply
appreciative of the considerable time and effort our contributors devoted to the preparation
of their articles.
Our expectation is that this Handbook will be very useful as a source of information and
inspiration to graduate students and young research workers who are entering the subject.
The material included will be of special interest to researchers in Banach space theory who
may not be aware of many of the beautiful and far reaching facets of the theory. We ourselves
were surprised by the new light thrown by the Handbook on directions with which
we were already basically familiar. We hope that the Handbook is also valuable for mathematicians
in related fields who are interested in learning the new directions, problems,
and methods in Banach space theory for the purpose of transferring ideas between Banach
space theory and other areas.
Our introductory article, "Basic concepts in the geometry of Banach spaces", is intended
to make the Handbook accessible to a wide audience of researchers and students. In this
chapter those concepts and results which appear in most aspects of the theory and which go
beyond material covered in most textbooks on functional and real analysis are presented
and explained. Some of the results are given with an outline of proof; virtually all are
proved in the books on Banach space theory referenced in the article. In principle, the
basic concepts article contains all the background needed for reading any other chapter in
the Handbook.
Each article past the basic concepts one is devoted to one specific direction of Banach
space theory or its applications. Each article contains a motivated introduction as well as

vi
Preface
an exposition of the main results, methods, and open problems in its specific direction.
Most have an extensive bibliography. Many articles, even the basic concepts one, contain
new proofs of known results as well as expositions of proofs which are hard to locate in
the literature or are only outlined in the original research papers.
The format of the chapters is as varied as the personal scientific styles and tastes of the
contributors. In our view this makes the Handbook more lively and attractive.
There are many strong and intricate interconnections between the different subjects
treated in the various articles, so some articles contain many cross references to other
articles. But, as mentioned above, no article depends on anything past the material treated
in the basic concepts article. Because of this lack of interdependence, we chose to order the
chapters after the basic concepts article alphabetically according to the first author of the
chapter. We included in volume 1 the chapters which came first in this order among those
articles which were ready by the deadline we set for this volume. The second volume of
this Handbook will contain the chapters which come later in the alphabetical order as well
as those chapters which were finished only after the deadline for volume 1.
This volume ends with a subject index and an author index for the material of the present
volume. Volume 2 will contain a subject and author index for the entire Handbook.
William B. Johnson and Joram Lindenstrauss

List of Contributors
Abramovich, Yu.A., Indiana University- Purdue University, Indianapolis, IN (Ch. 2)
Aliprantis, C.D., Purdue University, West Lafayette, IN (Ch. 2)
Alspach, D., Oklahoma State University, Stillwater, OK (Ch. 3)
Ball, K., University College London, London (Ch. 4)
Bourgain, J., Institute for Advanced Study, Princeton, NJ (Ch. 5)
Burkholder, D., University of lllinois, Urbana, IL (Ch. 6)
Casazza, E, University of Missouri, Columbia, MO (Ch. 7)
Davidson, K.R., University of Waterloo, Waterloo (Ch. 8)
Delbaen, E, E.T.H., Zurich (Ch. 9)
Deville, R., University of Bordeaux, Bordeaux (Ch. 10)
Diestel, J., Kent State University, Kent, OH (Ch. 11)
Dilworth, S.J., University of South Carolina, Columbia, SC (Ch. 12)
Enflo, E, Kent State University, Kent, OH (Ch. 13)
Figiel, T., Institute of Mathematics, Sopot (Ch. 14)
Fonf, V.E, Ben Gurion University, Beer-Sheva (Ch. 15)
Gamelin, T., The University of California, Los Angeles, CA (Ch. 16)
Ghoussoub, N., University of British Columbia, Vancouver (Ch. 10)
Giannopoulos, A.A., University of Crete, Heraklion (Ch. 17)
Godefroy, G., Universite Paris VI, Paris (Ch. 18)
Jarchow, H., University of Zurich, Zurich (Ch. 11)
Johnson, W.B., Texas A&M University, College Station, TX (Chs. 1, 19)
Kisliakov, S., POMI, Saint Petersburg (Chs. 16, 20)
Koldobsky, A., University of Missouri, Columbia, MO (Ch. 21)
K6nig, H., University ofKiel, Kiel (Chs. 21, 22)
Lindenstrauss, J., The Hebrew University of Jerusalem, Jerusalem (Chs. 1, 15)
Lomonosov, V. Kent State University, Kent, OH (Ch. 13)
Milman, V.D., Tel-Aviv University, Tel-Aviv (Ch. 17)
Odell, E.W., The University of Texas, Austin, TX (Ch. 3)
Phelps, R.R., University of Washington, Seattle, WA (Ch. 15)
Pietsch, A., Jena University, Jena (Ch. 11)
Schachermayer, W., Vienna University of Technology, Vienna (Ch. 9)
Schechtman, G., The Weizmann Institute of Science, Rehovot (Ch. 19)
Szarek, S., Case Western Reserve University, Cleveland, OH
and Universit~ Paris VI, Paris (Ch. 8)
Wojtaszczyk, E, Warsaw University, Warsaw (Ch. 14)
vii

CHAPTER 1
Basic Concepts in the Geometry of Banach Spaces
William B. Johnson*
Department of Mathematics, Texas A &M University, College Station, TX 77843, USA
E-mail: johnson @math.tamu.edu
Joram Lindenstrauss t
Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
E-mail: joram@math.huji.ac.il
Contents
1. Introduction .................................................. 3
2. Notations and special Banach spaces ..................................... 3
3. Bases ...................................................... 6
4. Classical spaces ................................................ 13
5. Banach lattices ................................................. 20
6. Geometry of the norm ............................................. 30
7. Analysis in Banach spaces .......................................... 35
8. Finite dimensional Banach spaces ...................................... 43
9. Local structure of infinite dimensional spaces ................................ 53
10. Some special classes of operators ...................................... 61
11. Interpolation .................................................. 74
12. List of symbols ................................................ 81
References ..................................................... 83
*Supported in part by NSF DMS-9623260 and DMS-9900185, Texas Advanced Research Program 010366-163,
and US-Israel Binational Science Foundation.
+ Supported in part by US-Israel Binational Science Foundation and by NSF DMS-9623260 as a participant in
the Workshop in Linear Analysis & Probability at Texas A&M University.
HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 1
Edited by William B. Johnson and Joram Lindenstrauss
9 2001 Elsevier Science B.V. All rights reserved

Basic concepts in the geometry of Banach spaces
1. Introduction
In this introductory chapter we present results and concepts which are often used in Banach
space theory and will be used in articles in this Handbook without further reference. The
material we treat, while familiar to experts in Banach space theory, has not made its way
into introductory courses in functional analysis. The main purpose of this article is to make
the subsequent articles accessible to anyone whose background includes basic graduate
courses in analysis and functional analysis. Each section of this article is devoted to one
aspect of Banach space theory.
Although this article can in no way be considered as an introductory course in Banach
space theory, we do include indications of proof of some basic results in the hope that
this will help the reader understand and get a feeling for the various concepts which are
discussed. We reference only some of the (mostly introductory) books which treat the basic
material we describe. Original sources are referenced in these books. We also mention
some results which are not yet in elementary books on Banach spaces but which help to
clarify the general picture. These generally were either discovered recently or are more
difficult than the rest of the material. In these cases we refer to specific articles in this
Handbook which treat the topic.
In general, we do not attach names to theorems (except when experts generally refer to
the theorem with a name attached, such as the Hahn-Banach theorem, the Krein-Milman
theorem, Rosenthal's s theorem .... ) or give any historical background. Instead we refer
to the books in the references as well as the articles in this Handbook.
2. Notations and special Banach spaces
Banach spaces will have either real or complex scalars. When the scalar field matters (for
example, in results involving spectral theory or in theorems of an isometric nature or when
analyticity plays a r61e), the scalar field is mentioned explicitly, but in the notation for
special spaces the scalars are not specified.
Operators between Banach spaces are bounded and linear. An invertible operator T is
called an isomorphism. Two norms on a vector space are called equivalent if the identity
operator on X (with the two given norms) is an isomorphism. If IIT l] = 1 = I1T-1 ]], T is
called an isometric isomorphism or simply an isometry and the domain and range of T are
said to be isometric. We write X ~ Y to denote that the spaces are isomorphic. To denote
isometry we use the equal sign. An isomorphism from a Banach space onto itself is called
an automorphism. A Banach space Y is said to be a quotient of the Banach space X if Y
is isometric to X/Z for some closed subspace Z of X. By the open mapping theorem, Y is
isomorphic to a quotient of X if there is an operator from X onto Y.
If X ~ Y, d(X, Y) denotes the Banach-Mazur distance between the spaces, defined
to be the infimum of IITII lIT -1 II as T ranges over all isomorphisms from X onto Y.
So d(X, Y) = 1 if X and Y are isometric; the converse is true for finite dimensional
spaces but not for infinite dimensional spaces. Note that the "triangle inequality" for the
Banach-Mazur distance is submultiplicative rather than subadditive; that is, d(X, Y)

- - Continuous
- - C
- The
W.B. Johnson and J. Lindenstrauss
A projection is an idempotent operator. A subspace Z of X is said to be complemented if
there is a projection from X onto Z. This is the case if and only if Z is closed and there is a
closed subspace W of X so that W A Z = {0} and X = W + Z; we then write X = W 9 Z
and say that X is the direct sum of W and Z. In this case W is isomorphic to X/Z by the
open mapping theorem.
We regard X as a subspace of X** under the canonical embedding. Note that there is
always a projection of norm one from X*** onto X* (restrict the functionals on X** to X).
An operator T :X ~ Y between Banach spaces is compact; respectively, weakly compact;
if the image T Bx of the unit ball Bx of X has compact; respectively, weakly com-
pact; closure in Y. If either X or Y is reflexive, then every operator from X to Y is weakly
compact. The identity operator on X is compact if and only if X is locally compact if and
only if X is finite dimensional.
A sequence {Xn}n~ in a Banach space X is called weakly Cauchy provided {X*(Xn)}n~=l
converges for every x* in X*. Identifying a Banach space with a subspace of its bidual, and
taking into account that the uniform boundedness principle implies that a weakly Cauchy
sequence is bounded, we see that {Xn }nO~ 1 is weakly Cauchy in X if and only if it converges
weak* in X**. The space X is said to be weakly sequentially complete provided every
weakly Cauchy sequence {Xn}neC__ 1 in X converges weakly, which is the same as saying that
the weak* limit of {Xn}n~176 1 in X** is in X itself.
Here is a list of special classical Banach spaces and other objects. The elementary theory
of these can be found in beginning texts in real and functional analysis. A more detailed
list of symbols, including some notation undefined in the text because we regard it as
"standard", is contained in Section 12.
N
R
C
T
C(K;X)
C(K)
Lp(lZ)
L~(#)
Lp(T)
~p(F)
ep
n
,ep
C
co(F)
-- The natural numbers.
= The real numbers.
= The complex numbers.
= The unit circle in the complex plane.
functions f on the (usually) compact Hausdorff space
K taking values in the (usually) Banach space X, normed by Ilfll =
suPtcg IIf(t)ll.
(K; X) when X is the scalar field.
-- The #-measurable functions f for which Ilfllp -- (f Ifl p d/z) 1/p < ~x~.
Here 0 < p < cxz.
#-measurable essentially bounded functions, with norm IIf I1~ --
infuz=0 sup Ifl~l. Lp(O, 1) -- Lp(#) when/z is Lebesgue measure on the
unit interval.
-- Lp (#) when # is normalized Lebesgue measure on the unit circle.
- Lp(lZ) when # is counting measure on the set F.
- ~p(F) when F = N.
--gp(/-') when F = {1, 2 ..... n}.
= The subspace of eoc of scalar sequences which have a limit.
- The closure in goc (F) of the scalar sequences which have finite support.

c0
Bx
Bx(x,r)
Basic concepts in the geometry of Banach spaces
= co(l-') when F = 1~.
-- The closed unit ball of the Banach space X.
= The closed ball of radius r with center x in the Banach space X; denoted
also B(x, r) when X is understood.
When 0 < p < 1, the space Lp(#) is not a Banach space except in the trivial cases that
it is zero or one dimensional. The metric on L p(#), 0 < p < 1, is given by p(x, y) =
IIx- yll p.
If {X~}n~__l is a sequence of Banach spaces and 1

W.B. Johnson and J. Lindenstrauss
product of the probability spaces (I-2n, I~'n) and for co = {con}nC~_ 1 in 1-2 set gn(co) = fn(con).
In particular, it is possible to define a sequence of independent random variables having
standard Gaussian distribution on the infinite product (0, 1)N of (0, 1) (with the product of
Lebesgue measure). By the isomorphism theorem for separable measure algebras [ 18, p.
399], (0, 1)r~ can be replaced by (0, 1) itself, but it is often more convenient to work on the
product space.
The characteristic function of the random variable g is the function ~0 :R --+ C defined
by ~0(t) = Ee itg. Useful algebraic identities include ~0(-t) = ~0(t); qgag+b(t ) -- e ibt~O(at);
and, especially, qgf+g = qgf~Og, valid when f and g are independent. The characteristic
function of a standard Gaussian random variable is e -t2/2.
A key fact is that two random variables (possibly defined on different probability spaces)
have the same distribution if and only if they have the same characteristic function. This is
proved by an inversion formula [ 10, 2.3.a], another simple consequence of which is that the
characteristic function of a random variable g is real valued if and only if g is symmetric;
that is, g and -g have the same distribution. A (by no means immediate) consequence of
the inversion formula [10, 2.7] is that for 0 < r < 2 the function e -Itlr is the characteristic
function of a (necessarily symmetric) random variable, called a symmetric r-stable random
variable. The tail distribution ~[Igl > t] of a symmetric r-stable random variable g is like
t -r as t ~ c~, so that Ilgllp < ~ for p < r, but Ilgllr - cx~.
Another probabilistic notion that plays an important role in Banach space theory is that
of martingale. First we recall the notion of conditional expectation. Let ~ be a probability
measure on a a-algebra/3. If ,4 is a sub cr-algebra of 13 and f is a ~-integrable function,
then by the Radon-Nikod3~m theorem there is a A-measurable function g so that for each
A in A, fA f dip -- fA g d~. The function g is called the conditional expectation of f given
A and is sometimes denoted by E(f I A) Suppose now that {fn}~ is a sequence of
9
F/=0
random variables on the same probability space and/3n is the smallest cr-algebra for which
fo, fl,..., fn are measurable. The sequence {fn }n~_0 is called a martingale provided that
for each n, fn -- E(fn+l I/3n). The sequence {dn}n~=0 of differences; do = fo, dn - fn -
fn-1; is then called a martingale difference sequence. Notice that a sequence {dn}n~=0
of independent random variables is a martingale difference sequence if and only if dn
has mean zero for all n ~> 1. A simple but important example of a martingale difference
sequence which is not a sequence of independent random variables is the Haar system,
discussed in Section 4. One basic theorem about martingales, the martingale convergence
theorem (see [10, 4.2.10]), states that every L1 bounded martingale converges a.e., which
means that if {fn }n=0 is a martingale of P-measurable functions and SUPn Elfnl < c~, then
{fn }n~__0 converges a.e. Moreover, if the martingale is uniformly integrable (see Section 4,
then it also converges in L1 (~) (see [10, 4.5.3]).
3. Bases
Excepting [2] and [12], which are oriented to Banach space theory, few introductory texts
in functional analysis treat Schauder bases. Nevertheless, bases are a very useful tool for
investigating properties of Banach spaces.

Basic concepts in the geometry of Banach spaces
We prove few statements in this section since most of the results are proved in [2] and
[12]. The book [14] often contains only sketches of proofs. Chapter 2 of [21] contains
enough details to be pleasant reading for the mature student or experienced mathematician
who is not an expert in Banach space theory. All of these books contain exercises. Many in
[21 ] are challenging even after peeking at the hints. Exercises in [ 14], as in this introductory
article, are scattered throughout the text and are sometimes flagged by such expressions as
"clearly", "hence", and the like.
A Schauder basis or simply a basis for a Banach space X is a sequence {Xn}n~=l of
vectors in X such that every vector in X has a unique representation of the form ~ OlnX n
with each C~n a scalar and where the sum is converges in the norm topology. The mapping
x w-> Otn then defines for each n a linear functional x n on X. One checks that the
expression !xV 9 = SUPn IIE~=I xk (x)x~ll defines a stronger complete norm on X, so that !.!
and II, II are equivalent by the open mapping theorem. One deduces from this that the
biorthogonal functionals for a basis are necessarily continuous. Moreover, the biorthogonal
functionals are a basic sequence in X*; that is, they form a basis for their closed linear
span. When it is useful to specify the biorthogonal functionals, we sometimes refer to the
"basis" {Xn, x n }n=l"
A sequence {Xn}n~=l is called normalized if each vector Xn has norm one and {Xn}n~__l is
called seminormalized if it is bounded and bounded away from zero. For most purposes it
is sufficient to consider normalized or at least seminormalized basic sequences9
The partial sum projections of a basis {Xn,'A n*l~ln__l ' defined for n = 1, 2,... by Pnx --
Y~.inl x*(x)xi, are necessarily uniformly bounded and converge strongly to the identity;
that is, Ilx - PnX II --+ 0 for each x in X. The supremum of the norms of these partial sum
projections is called the basis constant. This quantitative notion is of interest also if we just
consider finite basic sequences or bases for finite dimensional spaces. A sequence {x,},~__l
of nonzero vectors is basic with basis constant at most C if and only if for all n < m (and
all scalars O/i) the inequality II Y~in~__l OliXi ]l ~ C II Zi%l Oli Xi II holds. A block basis {yj }j= 1
of the basis {x~}~__l is a sequence of nonzero vectors of the form yj
m ~-'~nJ
z--~k=nj+l
+1
~kxk for
some sequence n l < n2 < .- .. The basis constant of a block basis of {Xn }n=~ 1 is no larger
than the basis constant of {xn}~
n=l"
A basis is called monotone provided that its basis constant is one. One can change to an
equivalent norm, Ill'Ill, in which the basis is monotone" just set ]llxlll- sup~ IlPnxll. In fact,
if one defines instead Illxlll = SUPn

W.B. Johnson and J. Lindenstrauss
you select Xn+l SO that maxx,cS [[Xn+l [[-1 [X*(Xn+l)[ is sufficiently small, the resulting sequence
{Xn}n~__l will have basis constant less than e + I-In~__l Cn. This yields, for example,
that every weakly null, non-norm null sequence has a basic subsequence.
This last result can be improved substantially. Two basic sequences {Xn }n~__l and {Yn }n~=l
are equivalent provided that the map Txn = Yn extends to an isomorphism from the closed
span of {Xn}nC~=l onto the closed span of {Yn}n~ K-equivalent if [[TllllT-1[[ ~< K (so,
strangely, {Xn}n~__l is 1-equivalent to {2Xn}n~__l, but usually basic sequences are normalized).
The principle of small perturbations says that if {Xn}n~__l is a basic sequence in
X and IlXn - Yn II--+ 0 sufficiently quickly, then {Yn}n~__l is a basic sequence which is
equivalent to {Xn}n~=l. To prove this, let {Xn*}n~__l be Hahn-Banach extensions of the functionals
biorthogonal to {Xn}n~__l to functionals in X* and define an operator S on X by
Sx -- Y~x*(x)(yn -Xn), SO [[S[[ ~ E [[x*[[[[yn --Xn[[. If I]sll < 1, elementary considerations
yield that I + S is an automorphism of X which maps Xn to Yn. This also yields
that if { n}n=l is a basis of X, then so is {Yn}n~_l. From this one sees that if {Xn}n~__l is
a basis for X with biorthogonal functionals tx n *~ J n=l' if {Yn}n~=l is a seminormalized sequence
in X, and if for each k, x k (Yn) -+ 0 as n --+ cx~, then {y~}n~__l has a subsequence
which is equivalent to some block basis {Zn}n~__l of {Xn}n~__l. From this it follows that every
infinite dimensional subspace Y of a Banach space X with a basis {Xn}n~__l contains, for
every K > 1, a normalized basic sequence {Zn}n~__l which is K-equivalent to a normalized
block basis {Yn }n~--_ 1 of {Xn}~__1. Moreover, the small perturbation argument indicated
above shows that the generated isomorphism which maps {Yn}n~=l into Y extends to an
automorphism on X.
To see how these basic facts about bases might be used, suppose that T is a noncompact
operator from a subspace X0 of a Banach space X with basis {Xn, x~ }n~__l into a Banach
space Y with basis {Zn}n= 1. Then there is a sequence {Yn}n~__l in the unit ball of X0 such
that for some E > 0 and all n ~ m, IITyn - Tym I[ > ~. By passing to a subsequence of
differences of {Yn}n~=l it can be assumed that for each k, x k (Yn) --+ 0 as n --+ ~. In view
of what was discussed in the previous paragraph, by passing to another subsequence it can
be assumed that {Yn}n~=l is an arbitrarily small perturbation of a block basis of {Xn}n~__l.
Repeating the same argument for {Tyn}n~=l in Y and normalizing at the end, we conclude
that if T is a noncompact operator from a subspace Xo of a Banach space X with basis
{Xn}n~__l into a Banach space Y with basis {Zn}n~=l, then there are automorphisms U on X
and V on Y and a normalized block basis {Yn}n~__l of {Xn}n~ with Yn E U-1Xo and such
that {VTUyn}n~__l is a seminormalized block basis of {Zn}n~__l.
This last mentioned result gives considerable information in cases where all block bases
of a basis can be characterized. The simplest examples of bases are provided by the unit
vector basis {en}n~__l for s 1

Basic concepts in the geometry of Banach spaces
We should mention that, in contrast to the case of gp and ~r, it generally is quite difficult
to determine whether two spaces are isomorphic even when both are presented concretely
as similar but different function spaces.
Another consequence of the principle of small perturbations is that g l has the Schur
property, which means that every weakly convergent sequence in ~ j is norm convergent.
Indeed, otherwise there would be a weakly null sequence {Xn }n~=l in g l which is bounded
away from zero. The sequence {Xn}n~=l is necessarily bounded, so the sequence {Xn}n~_l
would have a subsequence {Xnk }~=l which is equivalent to a block basis of the unit vector
basis of g 1 and thus equivalent to the unit vector basis of g 1. This is a contradiction because
the unit vector basis of ~l is not weakly null. An easy formal consequence of the fact that
g l has the Schur property is that every weakly Cauchy sequence in f j is norm convergent.
In particular, el is weakly sequentially complete.
The most natural basis for Lp(O, 1), 1 ~ p < oo, is the Haar system {hn}~_0, where
h0 = 1 [0,1), and for n - 2 j 4- k with j = 0, 1 .... and k = 0, 1 ..... 2 j - 1,
hn = l[k2-J,(2k+l)2-J-1 ) -- l[(2k+l)2-j-~,(k+l)2-j ).
It is easy to check that if 1 ~< p ~< oo, then for each n and all scalars Oti the inequality
II~,i=o~ihillp n

10 W.B. Johnson and J. Lindenstrauss
A sequence which is an unconditional basis for its closed linear span is said to be unconditionally
basic. Since a block basis of an unconditional basis is clearly unconditional (with
unconditional constant no larger than that of the basis itself), the perturbation principle described
above yields that every infinite dimensional subspace of a space with unconditional
basis contains an unconditionally basic sequence.
The unit vector bases for ep, 1 ~< p < cx~, and co are the simplest examples of unconditional
bases. For 1 < p < c~, the Haar system forms an unconditional basis for L p (0, 1).
The "modem" proof of this proceeds via martingale theory. The exact unconditional constant
of the Haar system in Lp(O, 1) is computed in [23]. In Section 8 we point out that the
trigonometric system is an unconditional basis in Lp(7~) only for p -- 2.
A basis {Xn}n~__ 1 for X is shrinking provided the linear span of the biorthogonal functionals
{Xn}n= * ~ 1 is (norm) dense in X*, which is to say that {Xn*}n~__l is a basis for X*. Notice
that a bounded shrinking basis necessarily converges weakly to zero. The basis {Xn}n~__l is
boundedly complete provided that whenever the sequence {y~in__ 101i Xi }nC~_ 1 is bounded, then
it is convergent. If a basis is shrinking, then its biorthogonal functionals are a boundedly
complete basis for X*. Conversely, if the biorthogonal functionals form a boundedly complete
basis for their closed linear span Y, then the basis is shrinking (and hence Y = X*).
Similarly, a basis is boundedly complete if and only if its biorthogonal functionals are a
shrinking basis for their closed linear span. If {Xn}n~__l is a boundedly complete basis for
X and Y is the normed closed linear span of the biorthogonal functionals {xn* }n=l' ~ the
shrinkingness of {Xn*}n~__l implies that the natural evaluation mapping from X into Y* is a
surjective isomorphism (which is easily seen to be an isometry if the basis is monotone).
Consequently, a space with a boundedly complete basis is isomorphic to a separable conjugate
space. From these facts it follows, in particular, that a Banach space X with a basis
is reflexive if and only if some (or every) basis for X is both shrinking and boundedly
complete.
If X has an unconditional basis, then some (or every) unconditional basis is boundedly
complete if and only if X has no subspace isomorphic to co, while some (or every) unconditional
basis is shrinking if and only if no (or no complemented) subspace of X is
isomorphic to el. Consequently, a space with unconditional basis is reflexive if and only if
no subspace is isomorphic to either co or to ~1 if and only if its second dual is separable.
In contrast, there is a nonreflexive space which has both a shrinking basis and a boundedly
complete basis which is of codimension one in its bidual, [ 14, 1.d.2].
While most of the separable spaces encountered in classical analysis have bases, many
do not have an unconditional basis; in particular, the spaces L1 (0, 1) and C[0, 1 ]. Indeed,
co does not embed isomorphically into L1 (0, 1) (for example, because L1 (0, 1) is weakly
sequentially complete and co is not; see Section 4), so any unconditional basis for L1 (0, 1)
would have to be boundedly complete and L 1(0, 1) would be isomorphic to a separable
conjugate space. But L 1(0, 1) does not even embed isomorphically into a separable conjugate
space (since if L 1(0, 1) C X* and for t E (0, 1) x~ is a weak* cluster point of the
sequence n -l l(t,t+l/n), then it can be checked that there is an uncountable subset S of
(0, 1) and 6 > 0 so that for all t :~ s in S, IIx? - x* II > 6). A more sophisticated (but actually
technically easier) argument shows even that L1 (0, 1) does not embed isomorphically
into a space with unconditional basis; see [14, 1.d.1]. Since, as mentioned in Section 4,

Basic concepts in the geometry of Banach spaces 11
every separable space embeds isometrically into C[0, 1 ], C[0, 1 ] also does not embed isomorphically
into a space with unconditional basis.
It is a deep result that there exist infinite dimensional spaces which do not contain an
unconditionally basic sequence (see [37]). The example uses Banach spaces whose norm
is not defined explicitly by a formula but by an implicit procedure. This method of defining
a Banach space (or a norm) is very useful in many contexts.
A basis {Xn}n~= 1 is called symmetric provided every permutation of {Xn}n~__ 1 is equivalent
to {Xn}n~176. In particular, every permutation of {Xn}n~__l is a basis, so a symmetric basis is
unconditional. A basis {Xn}n~__l is called subsymmetric provided it is unconditional and
equivalent to each subsequence of itself. A symmetric basis is subsymmetric [14, 3.a.3].
It is evident that if a basis is unconditional, symmetric, or subsymmetric, then the same is
true for the biorthogonal functionals (in their closed linear span).
The unit vector bases for ~p, 1 ~< p < ec, and co are symmetric, while the L p-
normalization of the Haar system is not a symmetric basis for L p (0, 1) if p 7~ 2. In fact,
L p(O, 1) has no subsymmetric basis if p 7~ 2. We have already mentioned that L 1(0, 1)
does not have even an unconditional basis. That L p(O, 1), 2 < p < ec, has no subsymmetric
basis follows from the dichotomy principle for sequences in the space, which is
discussed in Section 8. This principle says that if {xn}~~ 1 is a seminormalized basic sequence
in Lp(O, 1), 2 < p < ec, then {X~}nOC=l has a subsequence which is equivalent to
the unit vector basis for either s or g2. Since, as noted in Section 4, Lp(O, 1), p ~ 2, is
not isomorphic to either gp or ~2, Lp(O, 1), 2 < p < ec, has no subsymmetric basis. That
L p(0, 1), 1 < p < 2, has no subsymmetric basis then follows by duality.
It is interesting that symmetric bases are nevertheless prevalent. For example, any space
which has an unconditional basis is isomorphic to a complemented subspace of a space
which has a symmetric basis (this will be discussed further in Section 11). In fact, there
is even a space Y which has a symmetric basis and also an unconditional basis {Zn}n~
which is universal for unconditional bases in the sense that every unconditional basis for
any Banach space is equivalent to a subsequence of {Zn },c__ 1 ! The space Y and the universal
unconditional basis is easy to construct, given the fact, to be proved in Section 4, that
every separable Banach space is isometric to a subspace of C[0, 1]. Indeed, one takes a
dense sequence {X~}neC=l of nonzero vectors in C[0, 1] and lets Y be the completion of the
finitely supported scalar sequences under the norm II {an }~lll "- sup+ II Zn -+-anXn IIC[0,11.
The unit vector basis in Y is a universal unconditional basis. See [14, p. 129] or [41] for
further discussion and for the proof that the space Y has a symmetric basis.
While unconditional bases, symmetric bases, and subsymmetric bases are all useful
strengthenings of the notion of basis, there are equally useful weakenings. A Schauder
decomposition for a Banach space X is a sequence {X~ }n~__l of nonzero closed subspaces
of X such that every vector x in the space has a unique representation of the form x -- ~ Xn
with Xn ~ Xn. If each Xn is finite dimensional, the decomposition is called a finite dimensional
decomposition, or simply an FDD. A basis can be regarded as an FDD { Xn }~ 1 such
that every space Xn is one dimensional. As in the case of bases, a Schauder decomposition
{Xn}~~1761 for a Banach space X determines a sequence {Pn }~~176 1 (called the partial sum
projections of the decomposition) of commuting projections with increasing ranges which
converge strongly to the identity operator on X; namely, for x = y~ Xn with x~ E Xn and
n - 1, 2 ..... Pnx - ~-~inl xi. Conversely, if {Pn}n~__l is a sequence of commuting projec-

12 W.B. Johnson and J. Lindenstrauss
tions with increasing ranges which converge strongly to the identity operator on X, then
{(Pn - Pn-1)X}n~=l (P0 "-- 0) is a Schauder decomposition for X for which {Pn }n~_-i is the
sequence of partial sum projections. The biorthogonal functionals for a basis are replaced
in the case of a Schauder decomposition by the sequence {(P* - Pn*l)X* }n~=l of (even
weak*) closed subspaces of X*, which form a Schauder decomposition for their closed
span. The supremum of the norms of the partial sum projections determined by a Schauder
decomposition is finite and is called the decomposition constant of the decomposition, and
the Schauder decomposition is called monotone if its decomposition constant is one.
The definitions of unconditional, shrinking, and boundedly complete bases generalize
immediately to Schauder decompositions. The structure theory for bases goes over with
little difficulty to FDD's. More importantly, there are useful concepts involving FDD's
which are not merely generalizations of notions about bases. For example, a blocking of
a Schauder decomposition {Xn}n~=l is a sequence of the form {Xnk "-[-''" + Xnk+l-1}L1
with 1 = n 1 < n2 < .... Any blocking of a basis (that is, of a Schauder decomposition into
one dimensional spaces) produces an FDD which is not a basis. While bases and block
bases are fine in situations where passage to subsequences or subspaces involves no loss,
blockings of FDD's sometime provide the proper framework for investigations where it
is important to maintain global control. For example, it was mentioned above that every
normalized basic sequence in ep, 1 < p < cx~, has a subsequence equivalent to the unit
vector basis of ep, but one can prove that every FDD for a subspace of gp, 1 < p < ~, has
a blocking which is an ep decomposition [14, 2.d.1]. (A Schauder decomposition {Xn}n~__l
is said to be an g.p decomposition provided that for Xn ~ Xn, the series y~ Xn converges if
and only if y~ [[Xn[Ip < oo.)
A separable Banach space is said to have the bounded approximation property or simply
BAP provided there is a sequence { Tn }n~__l of finite rank operators on X which converges
strongly to the identity. If the operators can be chosen to commute, the space has the commuting
bounded approximation property (CBAP). In either case the supremum of [[ Tn [[ is
finite; if at most )~, we say that the space has the )~-BAP or )~-CBAP, or when )~ = 1 the metric
approximation property (MAP) or commuting metric approximation property (CMAP).
Finally, A Banach space X has the approximation property or simply AP provided that for
every compact subset K of X and E > 0, there is a finite rank operator T on X such that
[Ix - Tx l[ < ~ for every x ~ K. If a space X has the AP, then every compact operator S
into X is the norm limit of a sequence of finite rank operators. Indeed, take for n = 1, 2 ....
a finite rank operator Tn on X so that for each x in the image under of the unit ball of the
domain of S the inequality IIx - Tnxll

Basic concepts in the geometry of Banach spaces 13
conditions leads to nice theorems which do not involve any notion of approximation; in
particular, isomorphic characterizations of Hilbert space (see [35]).
The existence of spaces failing the AP as well as other considerations lead to the study
of other basis-like structures in Banach spaces. Probably the most useful of these is that of
Markuschevich basis. A Markuschevich basis for a Banach space X is a biorthogonal system
{x• x• *}• for which {x• }y~r is fundamental (that is, the span of the x• is dense
in X) and {X•215 * is total; that is, the x• *'s separate the points of X. The trigonometric
system in L I(~') or C(~') is a natural example of a Markuschevich basis which is not a
(Schauder) basis in any order (but this last statement is not easy to verify). Markuschevich
bases do not always exist in the nonseparable setting, but are a useful tool for investigating
nonseparable spaces in situations where they exist (see [44]). It is easy to see that
any separable space X admits a Markuschevich basis. One starts with linearly independent
sequences {Yn}nCX~ 1 in X and {'Yn*'CX~ln--I in X* with {Yn}n~ 1_ fundamental and {yn*}n~_l_ total
and applies a Gram-Schmidt type procedure to biorthogonalize the sequences (alternating
between working in X and X*) to produce a Markuschevich basis {Xn,X*}n~__l so that
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
fact [14, 1.f.4] is that a separable space contains a Markuschevich basis {Xn,Xn}n: * ~ 1 for
which SUPn IIxn II IIxn*ll ~< 20, and it is known that twenty can be replaced by any number
larger than one. We shall see in Section 8 that a finite dimensional space has a basis
, N
{Xn, x n }n=l for which IIxn II IIx* II - 1 for each n, but it is an open problem whether every
separable space has a Markuschevich basis with this property. One separable theorem in
which Markuschevich bases are used but do not appear in the statement is that every separable
space X has a subspace Y such that both Y and X~ Y have an FDD [ 14, 1.g.2]. Incidentally,
it is open whether the conclusion can be improved to "both Y and X~ Y have a basis".
4. Classical spaces
In this section we present some basic facts concerning the structure of the classical spaces
C(K) and Lp(#), their classification, and the relations among the spaces. Most books
on real analysis, such as [ 18], contain some elementary results about the structure of these
spaces. Beyond the most basic material, [21 ] is accessible to students and has the additional
advantage that some of the relations between these spaces and other spaces encountered in
analysis are touched on. The more narrowly focused book [7] is also directed at students,
contains several structural results omitted from [21], and the author's off-the-wall style
makes the book fun to read.
A normalized sequence {Xn}n~=l of disjointly supported functions in Lp(#), 1 0 there is a subspace Z of Y with
d (Z, X) < 1 § ~ and so that there is a projection of norm less than 1 + ~ from X onto Z.

14 W.B. Johnson and J. Lindenstrauss
Although general subspaces of X, X = s for 1 ~ p 7~ 2 < cc or X = co can be bad in
that they can fail the approximation property and there are many of them, in a sense that is
made precise in [24], the isomorphism theory of complemented subspaces of X is simple:
An infinite dimensional complemented subspace Y of X, X = g.p for 1

Basic concepts in the geometry of Banach spaces 15
Next, if Lp(l z) is separable with # a purely nonatomic probability measure, then the
isomorphism theorem for separable measure algebras [18, p. 399] yields that Lp(tX) is
isometric to L p(O, 1). On the other hand, if the measure # is purely atomic with atoms
{A• then the function which maps for each y the unit vector ey to lZ(Ay) -1/p times
the indicator function of A• extends to an isometry from g.p(F) onto Lp(lZ). Now if x
and y are in Lp(#), 1 ~ p ~ 2 < oc, then x and y are disjointly supported if and only if
Ilx + yll p = IlXllp + Ilyll p [18, p. 416]. This implies that an isometry from one Lp space
onto another preserves disjointness and therefore also atoms. All these remarks combine to
show that g.np, ~p, Lp(O, 1), s @p Lp(O, 1), ~pn @p Lp(O, 1), n -- 1, 2 .... , is a complete
listing, up to isometry, of the separable Lp(#) spaces when 1 ~< p 7~ 2 < co, and these are
all mutually nonisometric. Of course, in the Hilbertian case p - 2, s n -- 1, 2 .... ; ~2 is
the appropriate listing.
The decomposition method then yields that s s L p(O, 1) is a complete listing, up
to isomorphism, of the separable L p spaces. Later in this section it is noted that s is not
isomorphic to L p (0, 1) when 1 ~< p 7~ 2 < ec, so this is a listing of nonisomorphic spaces
for these values of p.
In order to study the structure of subspaces of L p (0, 1) as well as to investigate many
other questions about L p for one fixed value of p, it is convenient to consider the scale
of L p spaces as p varies. One reason for this is that (as we shall see) there are spaces of
functions X on (0, 1) on which the norm II" II p, is equivalent to II" II P2 with Pl < P2. Note
that when this occurs, since I1" II p ~< I1" IIr when p ~< r, all the norms I1" II p are equivalent
on X for pl ~< p ~< p2. In fact, the extrapolation principle says that then all the norms I1" II p
are equivalent on X for p ~< p2. Indeed, suppose C is a constant so that IIx II p2 ~< C IIx II pl
for x E X, 0 < p < pl, and 0 < )~ < 1 is defined by the formula pl = )~p + (1 - )Qp2.
Then, by H61der's inequality, for x E X we have
IIx II p~ ~ IIx II ~p IIx II (1-~)p= ~ C 1-~ IIx II p~l-& IIx II~p
and hence C 1-1/z IIx Ilpl ~ IIx lip ~ IIx IIp~.
The preceding yields particularly good information in case 2 < p and X is a closed
subspace of L p(O, 1) which is closed in Lr (0, 1) for some r ~: p. By the open mapping
theorem, the I1" II p and I1" II r norms are equivalent on X, and hence so is the I1" 112 norm. This
means that X must be isomorphic to a Hilbert space. Moreover, the orthogonal projection
P from L2 (0, 1) onto X induces a bounded linear projection/3 from L p (0, 1) onto X. The
extrapolation principle then says that I1" lip*, 1/p + I/p* = 1, and I1" 112 are equivalent on
X. The orthogonal projection onto X extends to the bounded projection 15, from Lp, (0, 1)
onto X.
Suppose that, on the other hand, X is a closed subspace of L p (0, 1), 1 ~< p < ~, and for
some (or, equivalently, every) 0 < r < p the I1" II p and I1" Ilr norms are not equivalent on
X. H61der's inequality then yields that for any M < cx~ the infimum of Ilxl[ixl

16 W.B. Johnson and J. Lindenstrauss
the principle of small perturbations, that X contains a subspace which is isomorphic to s
and complemented in L p (0, 1).
These observations yield a dichotomy principle for subspaces of L p(O, 1), 2 < p <
cx~ [21, III.A.4]. Let X be a closed subspace of L p(O, 1), 2 < p < oe. Then either X
is isomorphic to a Hilbert space and complemented in L p(O, 1) or, for each C > 1, X
contains a subspace C-isomorphic to s and C-complemented in L p (0, 1). This is stated
for Lp(O, 1), but is valid for general Lp(#) by the isometric classification of Lp spaces
discussed earlier. Notice also that the comments in Section 3 about s now yield that if s
embeds isomorphically into L p(#), 0 < r < cx~ and 2 < p < oo, then either r = p or
1~ --- 2.
The natural examples of infinite dimensional function spaces which are closed in L p
for more than one value of p come from probability theory. Suppose that {gn}n~=l is a
sequence of independent standard Gaussian random variables, discussed in Section 2. From
the form of the distribution of g it is evident that ]lg]lp is finite for all finite p and {gn}n~=l
is an orthonormal sequence in L2 (0, 1). A characteristic function argument yields that if
n 2 n
Y~=l [otk] -- 1, then Y~k= 1 ctk gk is again a standard Gaussian and hence has the same
norm in Lp(O, 1) as g. This means that for all 0 < p < oo, the mapping en ~ ][g][plgn,
n = 1, 2 ..... extends to an isometry from s onto a subspace X of L p (0, 1) where X does
not even depend on p, and the orthogonal projection onto X defines a bounded projection
from L p (0, 1) onto X if 1 < p < oe (in Section 10 we explain why no isomorph of s
in L 1(0, 1) or L~ (0, 1) can be complemented). In particular, the dichotomy principle for
subspaces of L p, 2 < p < oo, is not a monochotomy principle. This also implies, in view
of the structure theory of s discussed earlier, that s and L p (0, 1) are not isomorphic for
0 < p :/: 2 < o~.
A second natural example from probability theory of a subspace of L p, 0 < p < o0,
which is isomorphic (but not isometric for p # 2) to s comes from considering a sequence
{en}n~=l of independent random variables each taking on each of the values 1 and -1 with
probability 1/2. Such a sequence {en}n~__l is called a Rademacher sequence. Classically
the Rademacherfunctions {rn }n~__l are the concrete realization of such a sequence on [0, 1 ]
m V ~2n - 1
defined by rn " z_,~=o hz,+k, where {hn}n~__l is the Haar system. Khintchine's inequality
says that a Rademacher sequence is equivalent, in the Lp norm, 0 < p < o~, to the unit
vector basis of s for future reference we write the inequality explicitly with the best
constants labeled Ap and Bp:
Since a Rademacher sequence is orthonormal,
llz .ll
it follows that Ap =- 1 for p ~> 2 and Bp = 1 for p ~< 2, and the exact values of Ap and Bp
are known. The most elementary proof of Khintchine's inequality proceeds by checking
the right inequality in (1) for p an even integer. This clearly gives the result for all 2 ~<
p < oo and one then obtains the result for 0 < p < 2 from the extrapolation principle. The

Basic concepts in the geometry of Banach spaces 17
less computational modern proof of Khintchine's inequality gives a vector valued version
of Khintchine's inequality called the Kahane-Khintchine inequality. This will be done in
Section 8.
The existence of an r-stable variable g (see Section 2) for 0 < r < 2 shows that the subspace
structure of L p (0, 1), 0 < p < 2, is much more complicated than that of L p (0, 1),
2 < p < cx~. Indeed, just as in the Gaussian case, there exists for 0 < r < 2 a sequence
{gn}n~l of symmetric r-stable random variables defined on (0, 1). If 0 < p < r, then these
random variables are in Lp(O, 1). Now if ~k--1 n I~l r 1, then ~=l otkgk is again symmetric
r-stable and hence has the same norm in L p(O, 1) as g. This means that for all
0 < p < r, the mapping en ~ Ilgllp
-1
gn, n -- 1, 2 ..... extends to an isometry from er into
L p (0, 1). In Section 9 we explain how to derive from this the fact that for 1 ~< p < r < 2,
Lr (0, 1) embeds isometrically into L p(O, 1). It turns out that this covers all the cases in
which er embeds isomorphically into Lp(iZ ) with p and r finite. The remaining cases are
discussed in Section 8.
In the reflexive range 1 < p < cx~, information about subspaces of Lp,(#), 1/p +
1/p* - 1, given above gives information on quotients of Lp(#). It turns out that the case
p = 1 is quite different: Every separable Banach space X is isometric to a quotient of el.
Indeed, if {Xn}n~=l is dense in the unit ball of X, the linear extension of the map en ~ Xn
(where {en}~--1 is the unit vector basis for el) maps the unit ball of el onto a dense subset
of the unit ball of X and hence extends to a quotient mapping from el onto X. Similarly,
if the Banach space X has density character x, then X is isometric to a quotient of el (F)
when F has cardinality x.
Another useful and interesting property of el (which also holds for el(F) for any set
F) is the lifting property: If T is an operator from a Banach space X onto el then there
is a lifting S of T; that is, an operator S :el --+ X for which T S = Ie~. Indeed, by the open
mapping theorem there are Xn in X with Txn = en and )~ :=- supllxn II < cx~. The mapping
en ~ Xn then extends to an operator S:e 1 ---+ X with [[Sll - ;~ satisfying TS = le~.
As noted in Section 3, e 1 has the Schur property; that is, weakly convergent sequences in
el are norm convergent. An immediate consequence is that every weakly compact subset
of el is norm compact. The weakly compact subsets of L 1(0, 1) are more complicated
since L 1(0, 1) contains infinite dimensional reflexive subspaces. There is however a nice
characterization of subsets of L1 (/z) which have weakly compact closure when /z is a
finite measure. First, if X is any Banach space and W is a subset of X such that for each
E > 0 there exists a weakly compact set S so that W C S + EBx, then W has weakly
compact closure (use the fact that a bounded subset of X is weakly compact if its weak*
closure in X** is a subset of X). Next, given W C L1 (#) with # a finite measure, set
for k E N, ak- ak(W) "--sup{llxllxl~klll" x ~ W}. Clearly {ak}~= 1 decreases to some
a = a(W) ~ O. The set W is called uniformly integrable if a = 0. If a = 0 then W has
weakly compact closure because W is a subset of kBL~(~) + akBLl(~) and BL~(u) is
weakly compact in L1 (/z).
If a(W) > 0, the set W does not have weakly compact closure and in fact even contains
a sequence equivalent to the unit vector basis of el. To see this, define two further
numerical parameters for a subset W of L 1 (#); namely, b(W) := sup limn~ { Ilxn 1 An II1 },
where the supremum is over all sequences {Xn}n~__l in W and {An}~=l of measurable sets
with #An --+ 0; and c(W), defined the same way as b(W) except that the supremum is

18 W.B. Johnson and J. Lindenstrauss
over all sequences of disjoint measurable sets. It is an elementary exercise in measure
theory to verify that a(W) -- b(W) = c(W). Now if c(W) > 0, take a sequence {Xn}n~=l
in W and a sequence {An}n~=l of disjoint measurable sets so that 0 < [IXn 1an II1 ~ c(W).
Clearly c({xn 1Xn}) = 0, so {Xn 1X, } has weakly compact closure. The sequence {Xn 1 An }nCC_--I
is equivalent to the unit vector basis of e 1 since the An's are disjoint and IlXn 1An II is
bounded away from zero. This implies that a subsequence of {Xn}nCC=l is also equivalent
to the unit vector basis of el. (Remark: If {Yn}n~ is bounded in any L1 (IX) space
and there is a sequence {Bn}n~__l of disjoint measurable sets and b > 0 so that for all n,
Y-~k#n llynlsk 111 ~< b/2 < b

Basic concepts in the geometry of Banach spaces 19
as T and the range of T is separable because X~ Y is separable. Thus it is enough to check
that if Z is a separable space which contains co, then there is a projection of norm at most
two from Z onto co. To see this, first let zn * be an extension of the nth unit vector in ~ 1 = Co *
to a norm one element of Z*. Let d(., .) be a translation invariant metric on Z* which
induces the weak* topology on the unit ball of Z*; for example, if {Xn}n~ is dense in the
unit sphere of Z the metric can be defined by d(x*, y*) - y~n~__l 2-n I (x * - y*)(Xn)l. Since
every weak* limit point of {Zn*}n~ belongs to the unit ball B of the annihilator of co in Z*,
:r
it follows that d (Zn*, B) --+ 0. Thus we can choose w n * in B so that d (z*, w n) --+ O, which
means that Zn * -- W n * --+ 0 weak*. The formula Pz "-- {(z* - w,)(z)* }n=l ~ defines a projection
of norm at most two from Z onto co.
That the space co is not separably )~-injective for any )~ < 2 can be seen by considering
P 1 for any projection P from c onto co. More interesting is that the separability assumption
is needed: There is no projection from s onto co. Indeed, otherwise we would have
s ~ co (9 g~cc/co. But s is not isomorphic to a subspace of s This can be seen by
proving that c0(R) embeds into got/co but not into s c0(R) admits no countable separating
family of linear functionals, so it does not embed into s To see that c0(R) embeds
into s take a family {Ar}rER of infinite subsets of N so that the intersection of any
two is finite (replace N by the rationals and for each r E IR consider a sequence of rationals
which converges to r). Let Xr be the image in the quotient space s of the indicator
function of Ar. It is easy to check that {Xr}rEIR is 1-equivalent to the unit vector basis of
c0(R).
The space co is the only (up to isomorphism) separable space which is separably injective
[43]. General C (K) spaces do have another useful into extension property" A compact
operator T from a subspace Y of a Banach space X into C(K) has an extension T to
a compact operator from X into C(K). Indeed, since C(K) spaces have the BAR T can
be approximated in the operator norm by operators of finite rank and hence we can write
T = ~--~=0 oc Tn with each Tn of finite rank and II Zn II < 2 -n for n ~> 1. Therefore it is enough
to observe that if S'X --+ C(K) has finite rank, then S extends to a finite rank operator
S" X --+ C(K) with IISII ~< (1 + e)llSII (where e > 0 is arbitrary). But by using partitions
of unity and the principle of small perturbations one checks that there is a subspace E of
C (K) so that S X C E and d (E, g n) < 1 + ~, where n -- dim E < ec. Then E is (1 + e)-
injective and hence the desired extension of S exists. The same argument works when
C(K) is replaced by any/2oc space (defined in Section 9).
Every Loc(#) is isometric to C(K) for some compact Hausdorff space K. This follows
from Gelfand theory, since Loc(#) is a commutative B* algebra with unit (see [19,
Chapter 11]). Alternatively, it follows from lattice characterizations of C(K) spaces (see
Section 5).
The C (K) spaces play a special r61e in Banach space theory because they are a universal
class: Every Banach space X is isometric to a subspace of some C(K) space. This can be
seen by embedding X into goc (F) for F appropriately large and applying the comment in
the previous paragraph. Alternatively, X isometrically embeds via evaluation into C(K)
with K the unit ball of X* with the weak* topology. This approach is preferable because
the unit ball of X* is weak* metrizable when X is separable. In fact, every separable
Banach space X embeds isometrically into C[0, 1]. First, C(A) embeds isometrically into
C[0, 1 ], where A is the usual "middle thirds" Cantor set in [0, 1 ], by extending a continuous

20 W.B. Johnson and J. Lindenstrauss
function on A affinely on the component intervals of the complement of A in [0, 1 ]. Next,
using that A is homeomorphic to {0, 1 }r~, it can be shown that every compact metric space
K is the continuous image of A. Build a tree structure K(il, ie ..... in): {il, ie ..... in}
{0, 1}n; n = 1, 2 .... } of nonempty closed subsets of K so that each K(il, i2 ..... in) is
the union of K (il, ie ..... in, 0) and K (il, i2 ..... in, 1), K -- K0 U K1, and the maximum
diameter of K(il, i2 .... , in) as {il, ie ..... in} varies over {0, 1} n tends to zero as n --+ cx~.
This is not hard to do using compactness of K. The map defined by {ij}7-1 ~ K(il) N
K(il, i2) N K(il, ie, i3) N... is then a continuous surjection from A = {0, 1} N onto K.
Having gotten a continuous mapping g from A onto K, one embeds C(K) into C(A) by
the formula T x = x o g, x E C (K).
The isometric classification of C(K) spaces is easy and can be found in beginning texts
(such as [12, Theorem 231]): The space C(K) is isometric to C(H) if and only if K is
homeomorphic to H. This follows from the identification of the extreme points of the unit
ball of C(K)* as the evaluation functionals at points of K (as well as multiples of these
functionals by scalars of magnitude one). The much harder problem of the isomorphic classification
of C (K) spaces has been accomplished in the separable case (that is, for compact
metric spaces): If K is an uncountable compact metric space then C(K) is isomorphic to
C (0, 1). If K is a countable compact metric space then K is homeomorphic to the space
[ 1, or] of all ordinals up to the ordinal ot for some countable ordinal c~ in the order topology.
C (1, or) is isomorphic to C (1, t) when ot < fl if and only if fl < c( ~ See [40] for further
discussion of the separable case. It seems a hopeless task to get an isomorphic classification
of general C(K) spaces, but some information is contained in [44].
We have already mentioned inexplicitly that the dual of an L l(#) space is isometric to
a C (K) space. Similarly, the dual of a C (K) is isometric to L 1 (//~) for some measure/x.
The usual representation (see [18]) of the dual of C(K) is the space M(K) of finite signed
measures on the sigma algebra of Baire subsets of K (that is, the sigma algebra generated
by the closed G~ subsets of K). It is a routine exercise using the Radon-Nikod3)m theo-
rem to verify that if {/x• }•
is a maximal family of mutually singular Baire probability
measures on K, then M(K) is isometric to (Y-~yer L1 (/zy))l.
Once the duals of C(K) and of LI(/Z) are classified, it is natural to ask what are the
preduals of C(K) and L1 (/z)? It turns out that every isometric predual of a space C(K)
is isometric to L1 (/z) for some measure # and that all preduals of C (K) are mutually isometric.
On the other hand, the isomorphic preduals of, for example, g~, form a rich class
of spaces which are still not well understood. Preduals of L1 (/z) spaces are even less well
understood. Here we mention only that the space ~1 has uncountably many mutually nonisomorphic
preduals among the C(K) spaces (C(K)* is isometric to el if K is a countable
compact space).
5. Banach lattices
Most of the Banach spaces that appear naturally in analysis carry structure in addition
to their structure as Banach spaces. It tums out that this additional structure can affect
properties which are defined purely in terms of Banach space concepts (such as linear
operators and duality). An extra structure that classical spaces such as Lp and C (K) spaces

Basic concepts in the geometry of Banach spaces 21
possess is that of a Banach lattice, which is a Banach space over the reals that is equipped
with a partial order ~< for which x v y and x A y exist for all vectors x, y, and such that the
positive cone is closed under addition and multiplication by nonnegative real numbers and
the order is connected to the norm by the condition that Ixl ~< lYl =r Ilxll ~< IlYlI, where the
absolute value is defined by Ix l = x v (-x).
A linear mapping from a Banach lattice to a Banach lattice is positive if it carries positive
vectors to positive vectors. This is equivalent to saying that the mapping is order preserving.
It is easy to see that a positive linear mapping is continuous, so we call them positive
operators. If a positive operator preserves the lattice operations, it is called a lattice homomorphism.
It is clear that a one-to-one surjective positive operator T whose inverse is
positive is a lattice isomorphism; that is, both T and T-1 are lattice homomorphisms. The
dual of a Banach lattice X is again a Banach lattice under the standard ordering on X* in
which the positive linear functionals form the positive cone. With this definition it is easy
to see that the canonical mapping from X into X** is positive [ 15, 1.a.2].
While there are interesting functional analytical topics (such as positive operators and
lattice homomorphisms) which are special to Banach lattices, we are mostly interested
here in the Banach space properties of Banach lattices and concentrate on those lattice
properties which affect the linear topological or geometry of the underlying Banach space.
The basic reference for this aspect of Banach lattices is [ 15]. For a general introduction to
Banach lattices which covers the basic theory as well as some fairly recent material see [1 ].
The simplest examples of Banach lattices are the spaces with a monotonely unconditional
basis under the pointwise order on the coefficients, which as we have seen are better
behaved, or at least more regularly behaved, than general Banach spaces. It turns out that
much of the structure theory for these spaces carries over to Banach lattices.
It was already mentioned that the Lp and C(K) spaces are Banach lattices. Other examples
are the Orlicz spaces and the Lorentz spaces. An Orliczfunction is an even convex
function on R which is zero at zero and tends to infinity at infinity. If # is a measure and
M is an Orlicz function, the space LM(/Z) is the collection of all/z-measurable functions
f for which there exists C > 0 so that f M(f/C)d/z < ~. Then II f IIM is defined to be
the infimum of those C > 0 for which f M(f/C) d/z < 1. This is a norm on LM(/z) which
makes LM(/z) into a Banach lattice. If/z is counting measure on N, LM(/z) is called an
Orlicz sequence space and is denoted by gM. If 1 ~< p < ~ and W is a positive nonincreasing
continuous function on (0, ~) so that W(t) --+ ~ as t ~ 0+, W(t) --+ 0 as t --+ c~,
fl W(t) dt -- 1, and fo W(t) dt -- cx~, the Lorentz space LW, p(/z) is the space of all/z
measurable functions f for which IlflIw, p "-(f~xz f,(t)PW(t)dt)l/p ' where f* is the
decreasing rearrangement of Ifl. The space Lw, p(/z) is a Banach lattice under the norm
II f II W,p. Lorentz sequence spaces are defined in an analogous fashion.
The Orlicz spaces L M (/z) and the Lorentz spaces L w, p (/z), like the spaces Lp (/z), are
symmetric lattices which are ideals in the lattice of all/z-measurable functions. Here X is
symmetric means that if x E X and y is a/z-measurable function for which x* = y*, then y
is in X and IlYllx = Ilxllx. A subspace Y of a lattice X is an ideal provided x 6 X, y E Y,
and Ix l ~< l Yl =~ x E Y. Symmetric lattice ideals play an important role in interpolation
theory; see Section 11 for a short introduction. (Some sources call a symmetric lattice a
rearrangement invariant space; other references use "rearrangement invariant space" to

22 W.B. Johnson and J. Lindenstrauss
mean "symmetric lattice ideal", sometimes with extra conditions. That is why we avoid
the term "rearrangement invariant" in this article.)
It is no accident that in the examples of Banach lattices given above the lattice ordering
is just the natural pointwise a.e. ordering on a space of (equivalence classes of) real valued
functions on a measure space (for C(K) the measure is counting measure on K and for
spaces with unconditional basis the measure is counting measure on N). We call such a
Banach lattice a Banach lattice of functions or a Banach lattice of #-measurable functions
if it is important to specify the measure. There are representation theorems which say that
there is essentially no loss of generality in considering only Banach lattices of functions.
The most classical of these representation theorems, called the Kakutani representation
theorem, gives abstract Banach lattice characterizations of L1 (/z) and C (K) spaces. A Banach
lattice X is an abstract Lp space, 1 ~< p < cx~, provided IIx + yll p = Ilxll p + Ilyll p
whenever x and y are disjoint; that is, Ixl A lyl - 0. The Lp version of the Kakutani representation
theorem says that an abstract Lp space is lattice isometric to Lp(#) for some
measure lz (see [ 15, 1.b.2] or [ 1, Theorem 12.26]). Moreover, the measure/x can be chosen
to be a finite measure if the abstract L p space X has a weak order unit; that is, a vector
u ) 0 so that u A Ix l- 0 only when x is the zero vector. A Banach lattice X is an abstract
M space provided IIx + Yll - Ilxll v IlYll whenever x and y are disjoint. The C(K)
representation theorem says that an abstract M space is lattice isometric to a sublattice
of C(K) for some compact Hausdorff space K (see [15, 1.b.6]). Moreover, if the abstract
M space has a strong order unit (that is, a vector u ~> 0 such that the unit ball of X is the
order interval [-u, u] := {x: -u ~< x ~< u}), then M is lattice isometric to C(K) itself via
an isomorphism which maps u to Ilu II 1K.
Suppose that the Banach lattice X admits a strictly positive functional; that is, a positive
linear functional u* such that u*(Ixl) > 0 for every nonzero vector x in X. Define an
(inequivalent) norm I1" Ilu* on X by Ilxllu* := u*(Ixl). This is obviously an (incomplete)
abstract L1 norm on X from which it follows that the completion Xu, of (X, IIx Ilu*) is an
abstract L 1 space. Since the formal inclusion operator from X to Xu, is an injective lattice
homomorphism, the Kakutani representation theorem yields that X can be thought of as
a Banach lattice of functions. If X is separable, so is Xu,, and it follows from comments
made in Section 4 that Xu, is isometric and order equivalent to a sublattice of L l (0, 1), so
that X can be represented as a Banach lattice of Lebesgue measurable functions on the unit
interval.
Not every Banach lattice admits a strictly positive functional (co(F) with F uncountable
is a counterexample), but every separable Banach lattice X does. Indeed, take {Xn}n~=l
dense in the unit sphere of X and for each n pick a norm one functional x n * which achieves
its norm at Xn. It is easy to check that u* "- ~ 2 -n Ix~*l is a strictly positive functional.
From this comment and the representation above it follows that any lattice inequality involving
finitely many vectors which is true for lattices of functions must be true in a general
Banach lattice. This eliminates the tedium of verifying "obvious" inequalities (for example,
V( ~i=1 " ~:iXi 9 ~i -- -+'1} -~- ff-~i=l ]Xi[) directly from the axioms for a Banach lattice.
The representation of Banach lattices as lattices of functions suggests that abstract L1
spaces are particularly important lattices. The abstract M spaces also arise naturally in the
general theory. If u is a positive vector in a Banach lattice X and Xu is the linear span of
the order interval [-u, u] with [-u, u] taken as the unit ball, then Xu is easily seen to be

Basic concepts in the geometry of Banach spaces 23
an abstract M space with strong order unit u (Xu is complete because [-u, u] is closed in
the Banach space X).
A Banach lattice is order complete or Dedekind complete if every nonempty subset
which is bounded above has a least upper bound. A dual Banach lattice is order complete;
in fact, any norm bounded upwarded directed net in a dual Banach lattice converges
weak* to the least upper bound of the net.
Suppose that X is order complete and u ~> 0 with llu I] = 1. We look a bit more closely
at the C(K) space which is lattice isometric to the abstract M space Xu. Given x ~> 0,
define the support of x by S(x) := supn(nx)/x u. The supremum exists because X is
order complete. The support S(x) is a component of u. (A vector 0 ~< y ~< u is called a
component of u [or simply a component if u is understood] provided y is disjoint from
u - y.) If T:X, --+ C(K) is a lattice isometry with Tu = 1K, then Tx is an indicator
function if and only if x is a component. Of course, if A C K, 1 a is in C (K) if and only if
A is clopen (i.e., both open and closed). So the components form a Boolean algebra (a fact
which is also easy to verify directly) which is complete because X is order complete. Thus
the clopen subsets of K are also a complete Boolean algebra. An important fact is that the
clopen subsets of K form a base for the topology of K. One way to see this is to observe
that if 0 ~< x with x in Xu and t > 0 satisfies y := (x - tu) v 0 ~ 0, then 0 ~< tS(y) 0] contains a nonempty clopen subset. Since K is compact this implies that the clopen
subsets of K form a base for the topology. Using this and the completeness of the Boolean
algebra of clopen sets it is a simple exercise to prove that C (K) is itself an order complete
Banach lattice. As was pointed out in Section 4, this implies that C (K) is 1-injective.
From the discussion in the previous paragraph we can deduce: IfE is afinite dimensional
subspace of an order complete lattice X and ~ > O, then there is a finite dimensional sublattice
F of X and an automorphism T of X so that E C T F and ]l I - T ]l < e. Given the
subspace E, take positive vectors x l .... , xn in X whose span contains E and normalized
so that u := max/xi has norm one. Then F C Xu and Xu is isometric to a C(K) space for
which the clopen subsets of K form a base for the topology, which means that the span of
indicator functions of clopen sets is dense in C(K). Thus fixing a basis yl ..... yk for E
and 3 > 0, we get a subspace F of Xu spanned by disjoint vectors and a vector u in F with
]lull = 1 so that for each 1 ~< i ~< k, there is a vector Xi SO that ]Xi -- Zi] ~ (~U (and hence
Ilxi - zi [I ~< ~). Now apply the principle of small perturbations.
A Banach lattice is order continuous if every downward directed net whose greatest
lower bound is zero converges in norm (or weakly; it is the same) to zero. This is equivalent
to saying that every order bounded increasing sequence converges in norm (necessarily
to the least upper bound of the sequence) (see [15, 1.a.8] or [1, Theorem 12.9] or the
beginning of the argument below). It is also easy to check that an order continuous Banach
lattice is order complete [15, 1.a.8]. A Banach lattice is not order continuous if and only
if it contains a sequence of disjoint positive vectors which is equivalent to the unit vector
basis for co and is bounded above. The "if" direction is clear. If X is not order continuous,
one gets an upward directed net of positive vectors which is bounded above by, say, x,
with ]ix ]l = 1. The net cannot converge in norm, so one gets 0 ~< x l ~< x2 ~< ... ~< x so that

24 W.B. Johnson and J. Lindenstrauss
a := infn Ilxn+~ - Xn II > 0. Let Yn := Xn+l -- Xn ~ O. So we have for every n:
Yn >~ O, IlYn II ~ a, ~ Yk ~< x. (2)
k--1
(Although we do not need it here because we want to "disjointify" the yn'S, it is worth
noticing that (2) implies that Yn --+ 0 weakly and hence {Yn}n~__l has a basic subsequence,
and that any basic subsequence of {yn}n~__l is equivalent to the unit vector basis for co.)
Take Yn* in the unit sphere of X* so that Y*(Yn)= [[Yn[[. By replacing y~* with [Yn*[ if
necessary, we may assume that y~* ~> 0. For each n the nonnegative sum Y~m Y,~ (Ym) is at
most I[x [I. It is an elementary exercise in combinatorial reasoning to deduce from this that
for any E > 0 there is a subsequence {zk}~__l "-- {Ynk }~--1 of {yn}n~__l so that for each k > j,
Z k*(Zj) < 2-J6 2, where z k * "-- Ynk" * Thus for each n,
(Zn+l--6-1~Zk)k=l V0 Zn+ 1 Zn+ l -- 6 1 Zk >~ a - e.
k=l
This means that the zn's have big disjoint pieces. More precisely, let e = a/4 and set
(Z +l and Wn := (Vn -- EX) V O.
Then the Wn'S are pairwise disjoint positive vectors smaller than x with norms bounded
away from zero.
It is easy to see that {Wn}n~__l must be equivalent to the unit vector basis of co. This
completes the proof, but note that if X is order complete the mapping en ~ Wn from co
into X extends to a mapping from the positive cone of s to X by defining {~n6n }n~__l
sup n an Wn; this mapping extends to an order isomorphism from s into X.
Other useful characterizations of order continuous Banach lattices are given by the following
(see [15, 1.b.16] or [1, Theorem 12.9]): X is order continuous if and only if every
order interval is weakly compact if and only if X is an ideal in X**.
Since c is not order continuous but is isomorphic to the order continuous Banach lattice
co, there is not a linear topological characterization of order continuity. While it is only a
sufficient condition for order continuity of X that co not embed isomorphically into X, this
condition is only a bit too strong as arguments similar to those given above show that if co
embeds into X then it embeds as a sublattice (see [1, Theorem 14.12]).
The functional representation theorems for order continuous Banach lattices are stronger
and more useful than the representation already mentioned for Banach lattices which have
a strictly positive functional. Here we just indicate what is going on and refer to [15] for
details. However, the motivated reader is encouraged to work out the details for himself or
herself (partly because the discussion in [15] wanders unnecessarily). First, it is not hard to
show that an order continuous Banach lattice which has a weak order unit u also admits a
strictly positive functional u* (see [15, 1.b.15] or [1, 12.14]) and thus can be thought of as

Basic concepts in the geometry of Banach spaces 25
a space of integrable functions on a finite measure space (S-2,#) with u* (x) = f x d# for
x in X. One can assume that u* (u) = 1 = Ilu II and Ilu* II ~< 2. Since X is order continuous,
one has for each x ~> 0 in X that (nu)/x x converges to x as n --+ cx~, so that Xu is dense
in X. By replacing the underlying a-algebra with the smallest cr-algebra for which all
the functions in X are measurable, it can be assumed that X is dense in L1 (#). Then
necessarily u > 0 a.e. By replacing the measure # with u d# and functions f in X by f/u,
one can assume u -= 1. Now it is possible to recover the/z-measurable sets and the measure
#. From the density of X in L 1 (/Z) it follows that the indicator function of a measurable set
is an element x in Xu which is a component of u. From this it is essentially obvious that the
L~(#) norm on Xu agrees with its abstract M space norm. If we represent Xu as a C(K)
space, then the sets of positive/z measure are mapped onto the nonempty clopen subsets of
K in an obvious way so that we can represent X and L l(#) as function spaces on K and
transfer the measure # to K. We then get that C(K) C X C L1 (#) with both inclusions
having dense range. Moreover, C (K) = L~ (#) and every/z-measurable set is equal/z-a.e.
to a clopen subset of K. Finally, X is an ideal in L1 (/z) (use again order continuity and the
fact that X contains all indicator functions).
One consequence of this representation is: Let X be an order continuous Banach lattice
which has a weak order unit. A closed subspace Y of X either embeds into L 1 (/z) for some
measure/z or contains a normalized basic sequence which is equivalent to (even equal to
a small perturbation of) a disjoint sequence. For a proof see [ 15, 1.c.8] or modify the proof
of the dichotomy principle discussed in Section 4. By applying L1 theory discussed in
Section 4 one gets that a subspace of an order continuous Banach lattice is reflexive if and
only if it does not contain a subspace isomorphic to either s or co and that a nonreflexive
Banach lattice has a sublattice which is order isomorphic to g l or co. The representation
can also be used to prove that a Banach lattice which does not contain a copy of co must
be weakly sequentially complete [15, 1.c.4].
Although we have used here the representation theorem for abstract L 1 spaces, it should
be mentioned that since an abstract L1 space is necessarily order continuous (every normalized
disjoint sequence is obviously 1-equivalent to the unit vector basis of ~ 1), some of
the ideas (particularly building components in X) can be used to prove Kakutani's representation
of an abstract L 1 space which has a weak order unit as a space L 1 (/z) for some
probability measure/z.
Expressions such as (EnL1 ]Xn]P) 1/p, 1

26 W.B. Johnson and J. Lindenstrauss
By working in the abstract M space Xu with u "-- ~--~nL 1 Ixnl, the C(K) space case gives the
general case. We can then interpret scalar inequalities involving such expressions in general
lattices (rather than just in lattices of functions). For example, Khintchine's inequality,
discussed in Section 4, reads
A p ~ IXn I 2 ~< E 6nXn
n=l
n=l
Bp [Xn[ 2
n=l
1/2
(3)
Auxiliary lattices such as X (~p) can now be defined and the "obvious" dualities checked.
X (g2) is especially noteworthy, since it can be regarded as the complexification of X by
regarding ~2 as C and defining multiplication by complex scalars in the obvious way (see
[15, p. 43] for details).
Starting from the scalar identity Iotl~ 1-~ = infc>oOCl/~ + (1 -O)C~/(~ one
gets in a similar way an interpretation for the expression Ixl~ ~-~ and that
IIIxl~176 ~ Ilxll~ 1-~ (4)
Also, if 1 ~< p < q ~< cx~, 1/r "--O/p + (1 -O)/q with 0 < 0 < 1 and Ctl ..... OtN are
positive scalars, then for any collection x l ..... xu of vectors in a Banach lattice we have
that
otnlxnl r

Basic concepts in the geometry of Banach spaces 27
The smallest such M is denoted by M(P)(T). Clearly M(1)(T) -- IITII. Similarly, if for a
linear mapping T from a Banach lattice into a Banach space the inequality
(i IlTx. IIp
n=l
~M Ixnl p
n--I
1/p
(8)
always holds for some constant M, then T is called p-concave and the smallest such M
is denoted by M(p)(T). Clearly M(~)(T) = [[TI[. Evidently if T is p-convex [p-concave]
then it is bounded and M(P)(T) ~ [[TI[ [M(p)(T) ~ [[TI[]. It is not hard to check the identities
M(P)(T *) = M(p,)(T) and M(P)(T) -- M(p,)(T*), where 1/p + 1/p* = 1. As functions
of p, M(P)(T) is nondecreasing and M(p)(T) is nonincreasing (see [15, 1.d.5] or
write down a proof for Lp(lZ ) and see that the H61der type inequalities (4), (5) allow a
translation to the lattice setting).
A Banach lattice X is called p-convex (p-concave) if the identity operator Ix on X is p-
convex (p-concave) and we then define M(P)(X) := M(P)(Ix) and M(p)(X) :-- M(p)(Ix).
These constants are called the p-convexity and p-concavity constants of X. So X is p-
convex (p-concave) if and only if X* is p*-concave (p*-convex).
A p-convex and r-concave Banach lattice can be renormed with an equivalent lattice
norm so that the p-convexity and r-concavity constants are both one [ 15, 1.d.8]. In particular,
a lattice which is both p-convex and p-concave is lattice isomorphic to an abstract
Lp space.
If T : X --+ Y is a positive operator between Banach lattices it is easy to check (see [ 15,
1.d.9]) the inequality
(i ITx, I p
n=l

28 W.B. Johnson and J. Lindenstrauss
By p-concavity, this last quantity is dominated by
p) lip
M(p)(X)
M(p) (X) Bp Ixnl 2
1/2
This gives the equivalence of the norm of the square function of X l .... , Xn with a certain
Rademacher average:
(5)
p) 1/p
A1 Ixnl 2
n=l

Basic concepts in the geometry of Banach spaces 29
Let {y*}neC__l be the functionals in Y* biorthogonal to {Yn}nCC=l. By composing S with (contractive)
projections onto the span of initial segments of {Yn}oc
n=l' it can be assumed that
Y is finite dimensional, in which case {Yn*} is a monotonely unconditional basis for Y*.
To prove the right inequality in (12), given y~nN=l OtnYn, choose a norm one functional
ZnL 1/3nyn , in y, so that II~--~=l N ot~y~ II- Y~'~=l/~ N y,( ~ ~-~=l N ot, y~). Using (6) and the left
inequality in (10), we then get
N
• Otn Yn -- Z 13notnS* y* (Tyn)
n=l
n--1

30 W.B. Johnson and J. Lindenstrauss
unconditional basis for a complemented subspace of an L1 (/Z) space is equivalent to the
unit vector basis for g~l. Since C(0, 1) and L 1(0, 1) contain subspaces isomorphic to ~2
while co and ~1 do not, this gives another proof that neither C (0, 1) nor L 1(0, 1) has an
unconditional basis.
By using a p-convexification procedure, it is possible to build a scale of Banach lattices
starting with a lattice X in a manner analogous to how the scale of L p(/z) spaces
is constructed from L1 (/Z). For simplicity, we assume that X is a lattice of/z-measurable
functions (for the general case see [15, p. 53]). For 1 < p < oc, the p-convexification of
X is the space X (p) of all/z-measurable functions x for which IxlPsign(x) is in X. This is
easily seen to be a Banach lattice under the norm
. - It
1/p
ILx 9
The space X (p) is p-convex with M (p) (X (p)) -- 1. More generally, if X is r-convex and s-
concave, then X (p) is pr-convex and ps-concave with moduli M (pr) (X (p)) ~ M (r) (X) 1/p
and M(ps)(X (p)) O, 1 + tx*(y) = x*(x + ty)

- - p(r)
Basic concepts in the geometry of Banach spaces 31
By using the triangle inequality one checks that the function Ilx+tyll-Ilxllt is an increasing
function of t on (0, ec) and thus
Similarly,
x*(y) lim
t-+O+ -t
IIx II
Thus if these two limits coincide, the value of x* (y) is uniquely determined. On the other
hand, if these two limits differ, it follows from the Hahn-Banach theorem that for any X
satisfying
IIx - ty II - IIx II IIx + ty II - IIx II
lim
~ )~ 0 there exists 6x(e) = 3(E) > 0 so that
sup{ II (x + y)/21l Ilxll- Ilyll- 1; IIx - yll- ~} - 1 - 6(e).
The function 6x(e) is called the modulus of convexity of X. It is geometrically obvious (and
even true) that in the definition of 3(e), the equalities can be replaced by the inequalities
Ilxll ~< 1, Ilyll ~< 1, IIx - yll/> ~ without changing the value of 6(e).
One of the first theorems to relate the geometry of the norm to linear topological properties
is the following. A uniformly convex space is reflexive. In order to see this, we may
assume as well that X is separable. If X is not reflexive, then for any )~ > 0 there is x** in
the unit sphere of X** whose distance to X exceeds 1 - )~. Let A be a countable subset of
the unit sphere of X* which determines the norm of Y -- spanX U {x**}; that is, for each
y** in Y, Ily**ll - sup{y**(x*)" x* 6 A}. Let {Xn}neC__l be a sequence in the unit sphere of
X so that x*(xn) --+ x**(x*) for every x* in A. Then limn,m-+,c Ilxn + xmll -- 2, while for
every n, liminfm-+ec Ilxn -- Xm II ~> 1 - )~. This argument shows that for nonrettexive X,
~x(e) = 0 for every 0 < e < 1 and in particular that X is not uniformly convex.
A Banach space X is said to be uniformly smooth if the function
--sup/IIx + rYll + Ilx - ryll
px(r)
/ 2
- 1. IIx II - Ily II - 1 }
satisfies p(r) = o(r) as r --+ 0. Again one checks easily that a uniformly smooth Banach
space is reflexive.

32 W.B. Johnson and J. Lindenstrauss
There is a complete duality between uniform convexity and uniform smoothness. The
space X is uniformly convex if and only if X* is uniformly smooth. This follows from a
formula which connects the moduli 8x and Px*"
px, (r) -- sup{rE/2 - 8x(~)" 0 sup{rE/2 - 8x(~): 0 ~< ~ ~< 2}. Conversely, given r > 0 and x*,
y* in the unit sphere of X*, take x and y in the unit sphere of X so that (x* + ry*)(x) =
IIx* + rY*ll and (x* - ry*)(x) = IIx* - ry*ll (if one wishes to avoid reflexivity at this stage
one can choose x and y where the norms of the functionals are almost attained). Then
IIx* + ry*ll + IIx* - ry*ll ~< x*(x + y) + ry*(x - y)
IIx + yll + rllx - yll - 28x(~) + ~r,
where E - IIx - y II. This proves the reverse inequality.
The moduli of convexity and smoothness of a Hilbert space H can be easily computed
from the parallelogram identity:
pH(r)
E2 E 2
1 = )
4 8
-- v/1 + "r 2- 1 = r
T +
2
Hilbert spaces are the "most uniformly convex" and "most uniformly smooth" spaces in
the sense that if X is any space whose dimension is at least two, then 8/-/(E)/> 8x(E) and
p/-/(r) ~< px(r). For infinite dimensional X this is an immediate consequence of Dvoretzky's
theorem, to be discussed in Section 8, but a direct 2-dimensional geometrical argument
gives the general case.
The moduli of convexity and smoothness of the L p spaces can be computed exactly. It
is somewhat easier to describe their asymptotic behavior near zero, and this is what really
matters for Banach space theory. The result is
E 2
(p- 1)--~- + o(e2), if 1 < p ~< 2,
p2P '
if2~

Basic concepts in the geometry of Banach spaces 33
z'P jr_ O(rP),
p - _
IO L p ( r: ) -- 7:2
ifl < p~ 0 (see [15, If.l]).
Since it is easier to do analysis on a Banach space which has a norm with good geometric
properties than on a general space, it is important to know which Banach spaces can
be equivalently renormed so as to become strictly convex or smooth or uniformly convex
or uniformly smooth, and it is useful to have in these last cases moduli which are as good
as possible. The simplest, but nevertheless useful, renorming technique is as follows. Suppose
that T is an injective operator from a Banach space X into a strictly convex Banach
space Y. Then it is easy to check that IIIxlll := Ilxll + IITxll is an equivalent strictly convex
norm on X. Moreover, if T is an isomorphism and Y is uniformly convex, then II1" III is an
equivalent uniformly convex norm on X. Now if X is separable, then there is an injective
operator T from X into ~2, so Ilxlll = IIx II + IITx II is an equivalent strictly convex norm
on X. Also, there is an operator S from ~2 into X with dense range, so S* is an injective
operator from X* into ~ --~2 and IIx*l12- IIx*ll + IIS*x*ll defines an equivalent strictly
convex norm on X*. Since the adjoint operator S* is weak* to weak continuous, I1" 112 is
dual to a (necessarily smooth) norm I1" 112 on X. If X is smooth and T is an injective operator
from X into ~2 then (llx II 2 + II Zx II 2) ~/2 is an equivalent norm which is both strictly
convex and smooth. Hence every separable Banach space has an equivalent norm which
is both strictly convex and smooth.
For certain nonseparable spaces; in particular, ~ (F) with F uncountable (see [6, Chapter
II.7]), there may be no equivalent strictly convex or smooth norm.
If we are interested in obtaining a uniformly convex or smooth equivalent norm we
have to restrict attention to reflexive spaces. However, not every reflexive space can be so
OO /7 .
renormed. For example, if ~,=1 g l )2 had an equivalent uniformly convex norm II II and
I1" II, denotes the restriction of I1" Ilto the nth coordinate space ~, then the expression
IIIx III := lim IIx II, (where "lim" is interpreted to be a limit over some free ultrafilter on N or
a Banach limit or the limit along an appropriate subsequence) defines an equivalent norm
on the finitely supported vectors in el which extends uniquely to an equivalent uniformly
convex norm on ~1, but this is impossible.
There is a characterization of those spaces on which there is an equivalent uniformly
convex norm. These spaces, called superreflexive spaces, are discussed in Section 9. The
superreflexive spaces are also the class of spaces on which there is an equivalent uniformly
smooth norm. (These deep facts are discussed in [29].) If a space X has an equivalent
uniformly convex norm II1" III, then the equivalent uniformly convex norms are dense in the
metric space of equivalent norms (considered as bounded functions on the unit sphere of
X). Take, for example, I1" II +E II1"111. The uniformly convex equivalent norms also form a G~
set since those whose modulus of convexity at 1/n is positive forms an open set. Thus the
equivalent uniformly convex norms on X is a dense G~ in the space of equivalent norms
on X. Since, as we mentioned, X* also admits an equivalent uniformly convex norm, it

34 W.B. Johnson and J. Lindenstrauss
follows by duality that the equivalent uniformly smooth norms on X is also a dense G~ in
the space of equivalent norms on X, hence so is the family of equivalent norms which are
simultaneously uniformly convex and uniformly smooth.
In Section 8 it is pointed out that an infinite dimensional L p (#) space cannot be equivalently
renormed so as to have a better modulus of convexity or smoothness than that of the
natural norm.
There has long been a desire to describe reflexivity geometrically (that is, to show that a
space is reflexive if and only if there is an equivalent norm on the space that satisfies some
geometrical condition), and the notion of uniform convexity pushed in that direction. This
problem was recently solved (at least for separable spaces) and is discussed in [29].
Early in a first course in functional analysis a student learns that a Banach space is reflexive
if and only if its closed unit ball is weakly compact. There is a beautiful and useful
characterization of weak compactness, called James' theorem, which does not explicitly
involve the weak topology: A nonempty closed convex subset C of a Banach space X is
weakly compact if and only if every x* in X* attains its maximum on C. The only if part
is of course trivial. For the hard direction see [11, Theorem 79] or [26] for an accessible
proof when X is separable. This theorem says that on a nonreflexive space X there exist
linear functionals which do not attain their norm on the unit ball of X. Nevertheless, the
functionals which attain their norm on the unit ball is a rich set: The Bishop-Phelps theorem
says: Let C be a nonempty closed bounded subset of a Banach space X. Then the
functionals which attain their maximum on C is (norm) dense in X*. This theorem, which
is the starting point of the theory of optimization on Banach spaces, has many extensions
and applications (see [25]).
We outline the proof of the Bishop-Phelps theorem. First note that if f is a continuous
bounded function on a complete metric space (U, d), then there is, for every E > 0, a point
u0 in U so that f(u) fl for (x, t) in Kl and
u*(x) + at

Basic concepts in the geometry of Banach spaces 35
bounded closed convex set C in a complex Banach space X so that for every x* in X*, Ix*l
does not attain its maximum on C. This is discussed in [26].
We conclude this section by mentioning the geometric meaning of the Radon-Nikod3)m
property or RNP, an analytical concept that will be discussed in Section 7. A slice of
a closed bounded convex set C is a set of the form S(C,x*,~) = {x E C: x*(x) >1
SUpyec x*(y) - c~} with x* in X* and ot > 0. C is called dentable provided that for each
E > 0 there is a slice of C which has diameter smaller than ~. In [8, Th. V 3.9] and [3,
Th. 5.8] it is shown that X has the RNP if and only if every nonempty closed bounded
convex subset of X is dentable. Here we show how dentability can be used to derive other
geometric conditions, in particular, a version of the Krein-Milman theorem valid for noncompact
subsets of a space which has the RNE If X has the RNP, then every closed bounded
convex subset C of X is the closed convex hull of its extreme points. Recall that a face of a
convex set C is a nonempty (necessarily convex) subset F such that if )~x + (1 - ;~)y E F
with x, y in C and 0 < )~ < 1 then x and y are in F. Extreme points are faces consisting
of a single point. From the definition of slice it is obvious that if I(x* - Y*)(Y) I ~< 3 for all
y in C then S(C, y*, ~ - 23) C S(C, x*, ~) as long as ot > 23. We first show that C has
an extreme point. Take any slice S(C, x*, c~) of C whose diameter is less than one. By the
observation above and the Bishop-Phelps theorem, there is a y* arbitrarily near x* which
attains a maximum on C and so that the points P1 in C at which y* attains its maximum
is contained in the slice S(C, x*, ~) and thus has diameter less than one. Evidently P1 is
a closed face of C and of course is dentable since X has the RNE By induction we get a
sequence {Pn}~~ so that Pn+l is a face of Pn and diamPn ~< 1/n. Since a face of a face
is a face, all the Pn's are faces of C and their intersection is an extreme point of C which
is in the slice S(C, x*, ~). To finish the proof, let K be the closed convex hull of the extreme
points of C. If K were properly contained in C, then the separation theorem and the
Bishop-Phelps theorem would yield a functional x* which attains a maximum on C and so
that the set of points P in C at which x* attains its maximum is disjoint from K. By what
we have proved P has an extreme point which is afortiori an extreme point of C, which
contradicts the definition of K.
It is open whether this extreme point property for every nonempty closed bounded convex
subset of a space actually characterizes spaces with the RNE The statement about extreme
points is however valid in a stronger form which trivially implies that the dentability
condition is equivalent to the RNE A point x in C is called a exposed point of C if there is
x* in X* which attains its maximum on C exactly at the single point x. If also the diameter
of the slice S(C, x*, c~) tends to zero as ot --+ 0+, then x is called a strongly exposed point
of C. For example, every point on the unit sphere of a uniformly convex space is a strongly
exposed point of the unit ball. The stronger statement to which we alluded above is: Every
nonempty closed bounded convex subset of a space with the RNP is the closed convex hull
of its strongly exposed points.
7. Analysis in Banach spaces
In this section we describe the basic facts concerning integration and differentiation in
Banach spaces as well as the connection of these topics to convexity. Our "point of view"

36 W.B. Johnson and J. Lindenstrauss
is to take as known the scalar theory but not assume that the reader has any familiarity with
the vector valued theory. Much of what we treat in this section is contained either in [8] or
in [ 11,12]. Almost everything is in [3].
Let (s #) be a complete a-finite measure space and X a Banach space. A function
from s --~ X of the form ~in=l xi 1 ai with each Ai a measurable subset of s is called a
simple function. A function f : ~2 --+ X is called strongly measurable or just measurable
if it is the limit almost everywhere of a sequence of simple functions. A function f is
called scalarly measurable provided x*f is measurable for each x* in X*. A function
f is measurable if and only if it is scalarly measurable and the range of f is essentially
separable, which means that for some set A of measure zero, f[X2 ~ A] is separable. The
only if assertion is clear. To verify the if part, we may assume that X is separable and
#
select a sequence ~x*/cc in the unit ball of X* so that Ilxll - SUPn Xn (X) for every x in
t n "n=l
X. Thus for each fixed x in X, the function II f - x II is a measurable real valued function
on Y2. Since X is separable, given e > 0 there is a sequence {xn}~ec__ 1 in X and a sequence
{An}n~l of disjoint measurable subsets of s so that Ilf- ~nXnlAnll < ~. That is, f can
be approximated uniformly by functions taking only countably many values and having all
level sets measurable and hence f is the limit a.e. of a sequence of simple functions.
We next define the notion of the Bochner integral of a vector valued function. This is the
"strongest" of the various vector valued integrals which have been considered and is the
one most useful for the topics treated in this Handbook.
If f is a simple function supported on a set of finite measure, define f f d/x =
~x~X Iz[f = x]x. This is of course a finite sum. It is easy to see directly, and also follows
from the scalar case by composing with linear functionals, that the integral is linear
on this class of simple functions. If f is the a.e. limit of a sequence {fn }n~_-i of simple functions
supported on sets of finite measure and limn,m f IlL - fm II d# = 0, then f f d# is
defined to be limn f f~ d~. It is easy to check that f f d# does not depend on the particular
sequence {fn }n~_-i and that f f d# exists if and only if f is measurable and f Ilfll d# < oe.
That the integral has the expected properties either follows from the scalar case by composing
with linear functionals or is easy to check directly; see [8, pp. 44-52]. In particular,
II f f d# II ~< f II f II d# and T f f dlz = f Tf d# whenever f is integrable and T is an operator.
That the usual differentiability properties hold for the Bochner integral can be deduced
from the scalar theorems even without recalling the proofs in that case. Suppose that f is
a Lebesgue integrable X valued function on R n . Then for a.e. u in R n, we have
lim m(B(O, r)) -1
r--+O (u,r) Iif(v) - f(u)II dv = 0 (13)
and thus also f(u) - limr~om(B(O, r)) -1 fB(u,r) f(v) dv (here integration is with respect
to Lebesgue measure m on Rn). Indeed, we may assume that X is separable and {Xn}n~
is dense in X. Then by the scalar theorem,
lim m(B(O, r)) -1 s
r--+O
II f(v) - xnll dv = IIf(u) - Xnll
(u,r)

Basic concepts in the geometry of Banach spaces 37
for a.e. u and every n. For a point u for which this holds for every n we have
limsupm(B(O, r)) -1 fB
r--+O
(u,r)
fll
lif(v) - f(u)ll dv

38 W.B. Johnson and J. Lindenstrauss
Chapter 3]) is that a function f from the complex plane into Y is analytic if y* f is analytic
for each y* in Y*.
One important Banach space theory property that comes from differentiation theory is
the Radon-Nikodjm property or RNP. One way to define this notion is the following: The
space X has the RNP if every Lipschitz function from N into X is differentiable a.e. Here
we do not need to specify G~teaux or Fr6chet differentiability since the notions obviously
coincide for all functions from R. The RNP is usually defined by the requirement that
the Radon-Nikod~m theorem holds for X valued measures of finite total variation (see,
for example, [8]). We shall discuss that aspect and other equivalences of the RNP later in
this section. Here just note that a Banach space valued Lipschitz function f on [0, 1 ] has
separable range, so its a.e. derivative fl, if it exists, is also separably valued and thus is a
measurable function since x*f is measurable for each x* in X*.
First let us get a feeling for which spaces have the RNP. It is clear from the definition
that a subspace of a space with the RNP has the RNP and that a space all of whose
separable subspaces have the RNP has the RNP as well. The mapping t w-~ l(0,t) from
(0, 1) into L 1(0, 1) shows that L 1(0, 1) fails the RNP. Also co fails the RNP. This is seen
by considering the function f'R --+ co @~ co defined by f(t) - {fo sinnsds}n~--1 9
{fo cosns ds}n~__l . That f maps into co follows from the Riemann-Lebesgue lemma but
the only possible candidate for a derivative is {sinns}n~__l 9 {cosns}n~__l which is not in co
for any s.
Separable conjugate spaces have the RNP and thus all reflexive spaces have the RNP
(incidentally, this yields another proof of the fact mentioned in Section 3 that L 1 (0, 1) does
not embed into a separable conjugate space). To see this, let f be a Lipschitz function into
a separable conjugate space Z* and let {Zn }ne~=l be dense in Z. For all n, the scalar Lipschitz
function f(t)(Zn) is differentiable for a.e.t. At a point to where all of these functions are
differentiable, f(to)(z) is differentiable for every z in Z (observe that h-l(f(to -+- h) -
f (to))(z) - k -1 (f (to + k) - f (to))(z) --+ 0 as h, k --+ 0 because the difference quotient is
uniformly bounded since f is Lipschitz and tends to zero on the dense set {Zn }n~=l). From
this we conclude that the limit g(t) := limh__,oh-l(f(t + h) - f(t)) exists a.e., but the
limit is only in the weak* sense. This is all that can be said using just the separability of Z.
However, since Z* is separable, we deduce that g is measurable (g has separable range and
Jig(t) - z* II is clearly measurable for every z* in Z*; this is all that was used in the proof of
measurability in the beginning of this section). Also g is bounded by the Lipschitz constant
of f, so the Bochner integral G(s) "- fo g(t)dt is well defined for all s. Evaluating both
sides at an arbitrary z in Z we see from the scalar theory that G(s)(z) = (f(s) - f(0))(z)
so that f(s) -- G(s) + f(O) and hence by what we proved on the differentiation of the
integral we conclude that f'(s) = G'(s) = g(s) a.e. also in the sense that Ilh -1 (f(s + h) -
f(s)) - g(s)ll ~ 0 as h -+ 0.
Thus every subspace of a separable conjugate space has the RNP and a space with the
RNP has no subspace isomorphic to co or L 1(0, 1). These facts give useful criteria for
the RNP and in certain cases, such as spaces with an unconditional basis, allow a complete
determination whether a space has the RNR However, there do exist separable spaces
with the RNP which do not embed into a separable conjugate space and there are spaces
failing the RNP which do not contain isomorphic copies of either co or L 1(0, 1) (see [3,
Section 3.4]).

Basic concepts in the geometry of Banach spaces 39
Before stating the usual definition of the RNP we need to introduce the notion of what
is termed in [8] a "countably additive vector measure on a a-algebra". If X is a Banach
space, an X valued measure r is a countably additive function from a a-algebra into X. For
this to make sense, the series y~ r(A,) must converge unconditionally for each sequence
{An}n~__l of disjoint sets in the o--algebra. One defines the total variation It] of an X valued
measure by ]r ](A) = sup ~n lit (A,)11, where the supremum is over all partitions of A into
finitely many (or countably many; it is the same) disjoint sets in the cr-algebra (s The
measure r is said to be offinite variation provided ]r 1(s < cx~. If r is of finite variation,
then It] is a finite scalar measure. For example, if s = 1~,/3 is the collection of all subsets
of I~t, and {x,},~__l is a sequence in the Banach space X, then the assignment r{n} "- Xn
extends to an X valued measure on/3 if and only if ~n x. converges unconditionally. The
measure is then of finite variation if and only if ~.n ]]Xn 11 < c~. An example of an L p (0, 1),
1 ~< p < cxz, valued measure is gotten by taking the Lebesgue measurable subsets of [0, 1 ]
and defining r(A) = 1a. This measure has finite variation only if p = 1.
The theory of X valued measures and more general vector measures is exposed in [8].
In addition to their importance for the geometry of Banach spaces, vector measures are
centrally important for spectral theory; the ones encountered there typically do not have
finite variation.
An X valued measure r is absolutely continuous with respect to thefinite scalar measure
# provided r (A) = 0 for every set of # measure zero. Because # is finite, this is equivalent
to saying that for every ~ > 0 there is ~ > 0 so that Ilr(A)]l < ~ whenever #(A) < 6 (see
[8, p. 10]). If r is an X valued measure of finite variation it is clear that r is absolutely
continuous with respect to its total variation It] and even satisfies (14) below (with # :--
Irl).
An X valued measure r is differentiable with respect to a scalar measure lz provided that
there is an X valued measurable function g so that r(A) = fa g d# for every measurable set
A. We say that the Radon-Nikodym theorem holds in X provided that if r is an X valued
measure of finite variation and r is absolutely continuous with respect to a finite scalar
measure #, then r is differentiable with respect to #. If X satisfies this condition only for
all separable finite scalar measures, we say that the separable Radon-Nikodym theorem
holds in X (a measure # is called separable provided L1 (#) is separable). The usual
definition is that a Banach space X has the RNP provided the Radon-Nikodym theorem
holds in X and this is equivalent to saying that the separable Radon-Nikod3~m theorem
holds in X (see [8, Chapter III]). Later we prove this equivalence for separable X, but first
we show a general space X has the RNP if and only if the separable Radon-NikodSm
theorem holds in X.
Suppose that the separable Radon-Nikod3)m theorem holds in X and let f'[0, 1 ] ~ X
be a Lipschitz function. One defines a linear mapping T from the step functions on [0, 1]
into X by setting Tl[a,b] := f(b) - f(a) for a subinterval of [0, 1] and extending linearly.
Since f is a Lipschitz function, the mapping T is continuous when the step functions are
given the L 1(0, 1) norm, and hence T uniquely extends to an operator (also denoted by T)
from L1 (0, 1) into X. The assignment r(A) :-- T1A obviously defines an X valued measure
of finite variation which is absolutely continuous with respect to Lebesgue measure
m, so we get an X valued measurable function g on [0, 1] for which r(A) = fag dm for
every Lebesgue measurable subset of [0, 1 ]. In particular, f (t) -- f0 gdm + f (0) and thus

40 W.B. Johnson and J. Lindenstrauss
by what we proved in the beginning of this section f'(t) exists a.e. on [0, 1 ] (and is equal
to g(t)).
Suppose that X has the RNE To see that the separable Radon-Nikod3~m theorem holds
in X, suppose first that the scalar "control measure" Ix is Lebesgue measure on [0, 1] and
that the X valued measure r satisfies
[[r(A)I[ ~ 0 so that #(A) = fA f dv for every v-measurable set A. Of
course, IX is then also a separable measure and, as we have already remarked, r satisfies
(14), so from what we already have proved there is an X valued Ix-measurable function
g so that r(A) = fAg dix for every Ix-measurable set A. Then f.g is v-measurable and
r(A) = fA f" g dv for every v-measurable set A.
Observe that the simple argument reducing the study of an X valued measure which
is absolutely continuous with respect to a finite control measure to the case where the X
valued measure satisfies (14) yields another characterization of the RNP; more precisely,
that the (separable) Radon-Nikodym theorem holds in X if and only if for each operator T
from an L 1 (Ix) space with tx finite (and separable) into X there is an X valued measurable
function g so that Tf = f f g dix for all f in L I(IX). From this it is easy to see that if
X is separable and the separable Radon-Nikod3)m theorem holds in X then the Radon-
Nikod2~m theorem holds in X. Indeed, let IX be a finite measure on a cr-algebra B and let
T'LI(Ix) --+ X be an operator. Since X is separable there is a sequence {X*}n~ in X*
which separates the points of X. Let A be a countably generated sub ~r-algebra of 13 so
that all the Loc(Ix) functions T* x n * are A-measurable. Since {Xn}n=l , oc separates points of
X, Tf = TE(f I A) for each f in L1 (Ix). The restriction of Ix to A is a separable measure
since A is countably generated. Thus we get an X valued A-measurable function g so that
Tf = f f. g dIx for each A-measurable function f in L l (IX). But then for a general f in
L1 (Ix) we have Tf = TE(f I A) = f E(f I A)g dIx = f f. g dIx since g is A-measurable.
One of many places where the RNP arises naturally is in the study of vector valued L p
spaces. There is a natural isometric identification of Lp,(Ix, X*), 1/p + 1/p* = 1, with
a subspace of Lp(Ix, X)*, and for 1 ~ p < (x~, Lp(Ix, X)* = Lp,(Ix, X*) for allfinite (or
cr-finite) measures Ix if and only if X* has the RNP (see [8, Chapter IV]).
There are other important analytic characterizations of spaces with the RNP in terms of
martingales. In particular, the RNP spaces are exactly those Banach spaces in which the
martingale convergence theorem is valid in the sense that X has the RNP if and only if
every L1 bounded X valued martingale converges a.e. (see [8, Chapter V]).

Basic concepts in the geometry of Banach spaces 41
It turns out that in many places where one might assume reflexivity in order to use weak
compactness of the unit ball it suffices to assume that the space has the RNP.
We now discuss the differentiability of (real valued) convex continuous functions on a
Banach space X. Part of the importance of this topic derives from the fact that the norm
is a convex continuous function and differentiability of the norm is intrinsically related to
its smoothness. The most elementary reference for the differentiability of convex functions
and related topics is probably [11, Chapter 5].
An easy consequence of the definition is that a locally bounded convex function is continuous
and even locally Lipschitz. By using the separation theorem in X | IR it follows
that whenever f is convex and continuous in a neighborhood of a point x0 the set (called
the subdifferential off) Of(XO) :"- {X* E X*: x*(x -- xo) ~ f(x) -- f(xo) for all x 6 X} is
nonempty. From the theory of convex functions on R we know that for each u the right and
left derivatives of the function t ~ f (xo + tu) exist at t = 0. These one-sided derivatives
agree for every u (that is, all directional derivatives exist at x0) if and only if Of(xo) is
a single point which is then necessarily the G-derivative of f at x0. Consequently f is
G-differentiable at x0 if and only if for every u, f(xo + tu) + f(xo - tu) - 2f(x0) -- o(t)
as t --+ 0.
By considering f(x) = Ilxll we recover the fact mentioned in Section 6 that the norm
is G-differentiable at x0 in the unit sphere of X if and only if x0 is a smooth point of the
unit ball of X. It also follows that f is F-differentiable at x0 if and only if f(xo + u) +
f (xo - u) - 2f(x0) = o(llull) as Ilull -+ 0. If a convex function f is F-differentiable in a
neighborhood ofxo then D f (x) is continuous there; that is, F-differentiability of a convex
function on an open set implies that it is C 1 there. Indeed, suppose that Xn --+ xo and set
tO* "- Df(xo); u* "-- Df(xn). Given E > 0 there is 6 > 0 so that f(xo + y) - f(xo) -
w*(y) /(~/2)llu~* - w*ll
for all n. Then since u, * - Df(xn), we have by the convexity of f that
6 + tO* (Yn) + f (xo) >~ f (xo + y/7) >~ u* (Yn - Xn + xo) + f (Xn)
or
(~/2)llun* -- tO* II ~< (Un* -- tO* )(Yn)

42 W.B. Johnson and J. Lindenstrauss
The space X is uniformly smooth if the norm is uniformly F-differentiable on the unit
sphere; that is, if the limit limt~0 t -1 (llx + tull - Ilxll) exists uniformly in both x and u
on the unit sphere.
The classical G~teaux differentiability theorem for convex functions says: A continuous
convex function f on a separable Banach space X is G-differentiable on a dense G~
set. Indeed, if {Xn }n--1 ~ is dense in X then the set of G-differentiability of f is the set
('~n,m Gn,m, where Gn,m is the set of points x in X for which there exists 6 > 0 so that
f(x + 6Un) + f(x - gUn) - 2f(x) ~< 6/m. It is readily verified that each Gn,m is open and
dense.
For Fr6chet differentiability, the situation is much different even for norms. The norm of
el is G-differentiable at any point all of whose coordinates are nonzero but is nowhere F-
differentiable. This is typical in the sense that if X is separable and X* is nonseparable then
X admits an equivalent norm that is nowhere F-differentiable (see [11, Theorem 106]). On
the other hand: If X* is separable then every convex function f on X is F-differentiable
on a dense G~. That the set of points of F-differentiability is a G~ is easy to check (and
for this no separability assumption is needed); it is equal to ("In Gn where Gn is the set
of points x in X for which there exists 6 > 0 so that suPllull

Basic concepts in the geometry of Banach spaces 43
8. Finite dimensional Banach spaces
Since all finite dimensional spaces of the same dimension over the same scalar field are
mutually isomorphic, for results on finite dimensional spaces to be meaningful they must
be of a quantitative nature. The notion of Banach-Mazur distance is of central importance
in this context. Evaluating or even estimating the distance between spaces is often hard
since it generally is quite difficult to find an operator T for which IIT II II T-1II is minimal
or close to minimal.
We illustrate the computation of the Banach-Mazur distance by evaluating (or in some
cases just giving the order of magnitude) the quantity d (s s While relatively easy, even
in this simple situation it is by no means trivial to calculate or even closely estimate the
distance when p and r are on different sides of two. The topic of Banach-Mazur distances
between finite dimensional space is treated in many more involved situations in [20].
Denote the formal identity mapping from ~ to ~n by Ip,r. It is trivial to check that
IIIp,rll = 1 when p ~ r. Consequently, if 1 ~< p < r ~<
ec, then d(gp, ~) ~< n 1/p-1/r. The simplest and most important case occurs when p (or
r) is two. Either by induction from the case n -- 2, where the following equality is just the
parallelogram law, or by using the orthonormality of a Rademacher sequence and taking
the vectors xi to be in, e.g., L2(0, 1) and exchanging the order of integration over [0, 1]
with the expectation, one sees that vectors x l ..... x/7 in a Hilbert space satisfy the identity
~6iXi
i=1
2
- Ilxi ll2.
i--1
(15)
Suppose that 2 < r ~< cx~ and T's --+ ~ is an isomorphism normalized to sat-
isfy IIT-1II = 1 (so that IlTx[]2 /~ []Xl[r for all x). Denoting as usual the unit vectors
basis as {ei}, we see from (15) that n

44 W.B. Johnson and J. Lindenstrauss
By considering Vn* as the composition I2,1V*Icc,2 we see that IlVn*ll~,l ~< n. Since
Vn* = Vn 1 , it follows that d(g.~, gn)

Basic concepts in the geometry of Banach spaces 45
ot(T) Y
is defined by letting Tkei be xi when i --/= k and Tkek = x.
We turn to the proof of Lewis' lemma. Since the volume of T C for any measurable set
is a constant multiple of [det(T) Ivol(C), for any operator S:X ---> Y we have
det(To+S)
ot(r0 + S)
~< Idet(T0) I. (18)
Certainly To must be invertible, so by dividing (18) by Idet(T0)l we can rewrite (18) as
Idet(Ix + To'S) l

46 W.B. Johnson and J. Lindenstrauss
N ~< 2n2; see [20, 8.6].) Having done these preliminaries, we can state a nice geometric
reformulation of the case or(.) = I1" II in Lewis' lemma. Again identify both spaces X and
Y with ~n or C n. When one convex body contains another, a point x is called a contact
point of the bodies if it is in the intersection of their boundaries (so if the bodies are unit
balls associated with two norms, a contact point is a point in the intersection of the two
unit spheres).
Assume that Bx C By and vol(Bx) ~> vol(TBx) for every operator T on ~n [C n] for
which TBx C Br. Then there exist contactpoints xl ..... XN of Bx and Br and contact
points x 1 .... , x N of Bx, and Br', and ck ~ 0 so that I = ~=l ckx~ | xk. Also, N

Basic concepts in the geometry of Banach spaces 47
each n spaces Xn and Yn of dimension n so that infd(Xn, Yn)/n > 0. If one puts 2n pairs
of symmetric points on the unit sphere of ~, where the points are chosen independently
and are distributed uniformly on that sphere, and then takes the symmetric convex hull of
the union of these points with the unit vector basis, one obtains a unit ball for a (random)
space. If one takes two of these random spaces, then with big probability (that is, with
probability tending to one as n --+ cx~) the Banach-Mazur distance between the two spaces
exceeds gn for some constant g > 0 independent of n. Although this construction is easy to
describe, the computations are delicate and the reader is referred to [20, 38.1 ] for details.
The probabilistic approach has many other applications. For example, from the estimate
d(X, s 0 there exists no - no(e, k) so that if dim(X) ~ no then X contains a
subspace Y with d(Y, ~) < 1 + e. There are known good estimates on n0(e, k) (and even
better estimates for special classes of spaces). For general spaces n0(e, k) ~< exp(otk/e 2)
for some constant a. Dvoretzky's theorem is treated in detail in several books, including
[9, 19.1], [17, 4.3], [16, 1.5.8]. An exposition of Dvoretzky's theorem and related results
is given in [28]. There are many proofs of Dvoretzky's theorem and in most of them the
Dvoretzky-Rogers lemma is the first step.
From Dvoretzky's theorem and the technique for constructing basic sequences discussed
in Section 3 one gets in any infinite dimensional Banach space a basic sequence {Xn}n~=l
.2k+l
so that for each k, {xj }j__Zk§ 1 is 1/k-equivalent to an orthonormal basis in a Hilbert space.

48 W.B. Johnson and J. Lindenstrauss
An easy consequence of this is that for every square summable sequence {C~n}n~__l, of
scalars, every infinite dimensional Banach space contains an unconditionally convergent
series ~n Yn such that for each n, IlYn II = I~nl. One can also easily deduce this from the
simple Dvoretzky-Rogers lemma; in fact, historically it was this application that motivated
the discovery of the Dvoretzky-Rogers lemma.
A result related to Dvoretzky's theorem is Krivine's theorem:
If {Xn }n~__l is a basic sequence in a Banach space, then there exists 1 0 there is a
lattice isomorphism T from g n p into X with II T II II T-1 II < 1 + ~.
From Krivine's theorem and the fact discussed in Section 4 that ~2 embeds isometrically
into L p (0, 1) it is easy to deduce Dvoretzky's theorem. This is not the recommended
route to Dvoretzky's theorem as it is difficult to navigate through Krivine's theorem. More
importantly, the tight quantitative estimates obtainable from direct proofs of Dvoretzky's
theorem have many applications.
If we apply Krivine's theorem to a basic sequence which is equivalent to the unit vector
basis for ~r we see immediately that the only p that is obtainable is p = r. Thus by applying
Krivine's theorem to a disjoint sequence in an infinite dimensional L p (#) space we infer
that any equivalent renorming of Lp(#) contains for every k and ~ > 0 a subspace whose
Banach-Mazur distance to g k p is less than 1 + E This implies that an infinite dimensional
Lp(#) space cannot be given an equivalent norm which has a better modulus of convexity
or smoothness than its natural norm.
Two notions that are very important for both the finite dimensional and infinite dimensional
theories are that of type and cotype. A Banach space X is said to have type p provided
there is a constant C so that for every sequence x l ..... xn in X,
E • 6iXi
i=1

Basic concepts in the geometry of Banach spaces 49
a constant C so that for every sequence X l ..... Xn in X,
[[xill q
i=1
2) 1/2
The best constants in (22) and (23) are denoted by Tp (X) and Cq (X). By the parallelogram
identity characterization of Hilbert space, a space X is isometric to a Hilbert space if and
only if Tz(X) = 1 = C2(X). Even a one dimensional space does not have type p for any
p > 2 or cotype q for any q < 2. For every space X, 7"1 (X) = 1 and Coo(X) -- 1 by
convexity of the norm. As functions of p and q, Tp(X) is nondecreasing and Cp(X) is
nonincreasing. The inequalities Tp(X)

50 W.B. Johnson and J. Lindenstrauss
sional L~(#) spaces are universal for separable spaces, they do not have type p for any
p > 1. Similarly, an infinite dimensional Lq (#) space for 2 ~< q ~< oo does not have cotype
smaller than q.
If {Xn }n=l ~ is a seminormalized unconditionally basic sequence in Lq(lZ), 2 < q < oo,
with # a probability measure and inf IlXn 112 > 0, then since Lq(#) has type 2 and L2(#) has
cotype 2 and I1" 112 ~ I1" IIq we infer that {Xn}n~=l is, in Lq(#), equivalent to the unit vector
basis of g2. Thus such a sequence cannot have dense linear span in Lq(#). In particular,
the trigonometric system is not unconditional in Lq (0, 1) for 2 < q ~< c~ and, by duality,
also not in L p(O, 1), 1

Basic concepts in the geometry of Banach spaces 51
One can assume that I[x + YI[ + [Ix - YI[ - 2 with t "- ]]x+yll-llx-y[] ~ O. Then for e -- +1,
2
1 ~Crp 1 - Crp
Ilx + opey II ~< IIx + eyl[ + 2 I[x - eyl[- 1 + eCrpt.
Therefore
(EII + + ,/1 + t2- (EII + lyl12)
Finally, iterate (27) to obtain (25). Formally, for k -- 1 ..... n define Sk "-- ~ki= 1 6iXi; we
need to show that (EIIx + o'pSkllP) 1/p

52 W.B. Johnson and J. Lindenstrauss
arose first in the study of probability in Banach spaces and are sometimes called B-convex
(see [34]).
Similarly, a space does not have finite cotype if and only if it contains for every n arbi-
trarily close copies of s Since ek 1 is a subspace of ~ (isometrically in the case of real
scalars and up to constant in the complex case), we deduce that if X is B-convex then it
has cotype q for some q < cx~. Since a uniformly convex space cannot have subspaces of
arbitrary dimension which are arbitrarily close to s it follows that a uniformly convex
space is B-convex; that is, has nontrivial type and cotype.
The relation of B-convexity to reflexivity is not so simple but has been clarified. A B-
convex space need not be reflexive-there is even a nonreflexive space of type 2. Though
nonreflexive Banach spaces need not contain almost isometric copies of s they do contain
configurations of the following type for every n and ~ > 0: norm one vectors x i .... , Xn
so that for every k < n, Ilxl +"-+ Xk - (Xk+l + '''-~- xn)ll >~ n - E. In particular, taking
n : 2 we see that: Every nonreflexive space contains real subspaces arbitrarily close
to real s That is, a real space whose unit ball does not have a two dimensional section
arbitrarily close to a square must be reflexive. See [2, 4.111] for proofs of these results.
We next discuss the duality theory of type and cotype. It is simple that if X has type p
then X* has cotype p* and Cp,(X*)

Basic concepts in the geometry of Banach spaces 53
would be bounded, where P is the orthogonal projection onto the closed linear span of the
Rademacher sequence {en}~_l (a space X for which P is bounded is said to be K-convex).
/5 is defined explicitly for f/ in L2(#) and xi in X by ['(~i~=l fixi) -- Y~"i=l (P~)xi" It
is a deep fact (see [16, II.14], [17, 2.4], and [38]) that: X is K-convex if and only if X is
B-convex. Thus the presence of copies of ~ is exactly the factor which hinders a clean
duality between type and cotype. If X is B-convex and of cotype p* then X* is of type p.
For finite dimensional X it is important to estimate the norm of the Rademacher projection
(called the K-convexity constant of X and usually denoted by K(X)). A useful
estimate (which is valid also for isomorphs of infinite dimensional Hilbert space) is
K(X) 1, the distance estimate from John's theorem
gives that K (X) ~< K l log n. These estimates are sharp up to the values of the constants K
and Kj. For a discussion see [20, pp. 86-92], [17], [16, II.14], or [38].
9. Local structure of infinite dimensional spaces
In this section we describe results and techniques whose purpose is to relate the structure of
an infinite dimensional Banach space with the structure of its finite dimensional subspaces.
We will be particularly interested in properties of a space which depend only on its family
of finite dimensional subspaces and not on the way these finite dimensional spaces are
"glued together" to form the infinite dimensional space.
A Banach space X is said to be L-representable in a Banach space Y if for every finite
dimensional subspace E of X and every ~ > 0 there is a finite dimensional subspace F of
Y with d(E, F)

54 W.B. Johnson and J. Lindenstrauss
so that if (1 + 3)-l ~< II Sy II ~ 1 + 6 for every y in a 6-net in the unit sphere of Y, and some
operator S, then (1 + ~)-l [ly II ~< IlSy II ~< (1 + ~)IlY II for all y E Y.
We pass to the proof of the principle of local reflexivity.
Let e > 0 and let 6 = 6(e) be as above. Let E and F be finite dimensional subspaces
of X**, respectively, X*. We pick {ijj}j__ 1, m in the unit ball of X* so that the set contains
an algebraic basis of F and so that IIx**ll ~< (1 + 6)maxj Ix**(v~)l for every x** E E.
Let {wi**}/n__l be a 6-net in the unit sphere of E so that {w**}~ 1 is a basis of E N X and
{//)i**}i=1 r is a basis of E for some k ~< r < n
"
Write w i **
=Zh=l)~ihW
r h ** for r < i ~ n.
For r < i ~< n put #i,h -- Xi,h if h ~ r, ~ii = -- 1, and #i,h = 0 for r < h # i. Consider the
operator
/7
n-r+k
s. (2 e . . . 9 x)~ --, (x ~ i . . 9 x)~
d~
defined by
( )
S(xl ..... Xn)-- x1 ..... Xk, #r+l,hXh ..... #n,hXh 9
h-1 h-1
By the choice of #i,h for h > r it follows that S is onto. Hence the operator
/7
n-r+h
~. (x e . . . 9 x)~ --, (x e . . . 9 :~ 9 R'm)~
defined by S(xl ..... Xn) -- (S(xl ..... Xn), vj(xh)), 1 ~ j

Basic concepts in the geometry of Banach spaces 55
Recall that a family b/of subsets of a set I is called a filter if it is closed under finite
intersections, does not contain the empty set, and whenever A C B with A E b/then B
b/. A maximal (with respect to inclusion) filter is called an ultrafilter. By Zorn's lemma
every filter is contained in an ultrafilter. An ultrafilter is called free (or nontrivial) if the
intersection of all sets in b/is empty. An indexed family {Xi}i6I in a topological space
is said to converge to x with respect to a filter bt (in symbols, x -- lin~ xi ) provided for
every open set G containing x the set {i: xi ~ G} belongs to/g. A Hausdorff space is
compact if and only if every indexed family {xi }i~I converges (to a unique point) for every
free ultrafilter b/on I. Assume now that I is a set and b/is a free ultrafilter on I; assume
also that for all i, Xi is a Banach space. We define a seminorm Ill'Ill on (Y-~i Xi)ec by
IIIxlll = lirr~ Ilxi II where x -- {xi }i~/with xi ~ Xi for all i. The limit exists since a closed
bounded interval on the line is compact. The quotient of (~-~i Xi)ec with respect to the
closed subspace of all x with ]llx Ill --- 0 with its obvious norm is a Banach space, called the
ultraproduct of the Xi (with respect to b/), and is denoted by (1--Ii Xi)cr If all the Xi are
the same space X we call the space thus obtained an ultrapower of X, denoted also by Xu.
Ultraproducts of Banach spaces are treated in detail in [9, Chapter 8].
Given two families {Xi}icI and {Yi}i~I of spaces and operators 7):Xi ~ Yi with
sup/ ]]Ti ]] < cx:), there is a natural operator T:(1-Ii Xi)lg -'+ (Hi Yi)bt called the ultraproduct
ofthe operators 7). It maps an element in (1-Ii xi)cr represented by x = {xi}i6I in
(~_~ Xi)~ into the element in (]-Ii Yi)Cr represented by y = {~xi}is/.
The ultraproduct of one dimensional spaces is one dimensional and more generally if
dim Xi = n < ec for all i then (I-Ii xi)cr is also n-dimensional. On the other hand if I = N
and limu (dim Xi) = ec then (I-Ii xi)u is already nonseparable.
The ultraproduct of Banach lattices is again a Banach lattice if we take as the positive
cone in (Hi xi)cr the set of all elements which have representatives x = {xi}iEl in
(Y]~i Xi)oc with xi ~ 0 for all i. If all the Xi are abstract Lp spaces for some fixed p,
1 ~< p < ec, then (Hi xi)u is again an abstract Lp space and hence is isometric to Lp(#)
for some measure # by the L p version of the Kakutani representation theorem. Similarly,
if all the Xi are C (Ki) spaces for some compact Hausdorff Ki then so is (I-Ii xi)cr However,
for other families of Banach spaces Xi (even, e.g., if all are Orlicz spaces with the
same Orlicz function q)) the determination of the nature of (1-[i Xi)u is not an easy task.
As a first application of ultraproducts we shall prove now a fact mentioned already in
Section 4: If 1

56 W.B. Johnson and J. Lindenstrauss
(and in particular a reflexive space) there is a contractive projection from Xu onto the
canonical image of X in it. Map {Xi}iEI to w*-lingr xi. As for duality, the space (I-Ii x*)u
can be in a natural way identified with a subspace (usually proper) of the dual of (1--Ii xi)ct.
We now discuss the relation between ultraproducts and )~-representability. It follows
directly from the definitions that any ultrapower Xct of a Banach space X is finitely representable
in X. There is also a converse, in a sense, to this statement. A Banach space
X is ,k-representable in Y if and only if there is a subspace Z of some ultraproduct of Y
so that d(X, Z) 0. Introduce a partial order on I by
(El, 61) < (E2, 62) if E1 C E2 and 61 > 62, and let///be an ultrafilter on I containing for
all i E I the set {j E I: i < j}. By assumption there is for every (E, 6) 6 1 an operator
TE,~ from E into Y so that IIx II ~< II TE,~x II ~< ()~ + 6)Ilxll for all x E E. For every x 6 X
let~ = {xE,~} 6 (Y~'~i Y)~ be defined by xE,~ = TE,~x ifx 6 E and xE,~ =0ifx ~ E. The
image of all these s in YU is easily seen to be a subspace Z which satisfies d (X, Z) ~< )~.
From the principle of local reflexivity it follows that for every Banach space X the
space X** is isometric to a norm one complemented subspace of a suitable ultraproduct
of X. Without the complementation assertion this is a special case of the result above.
To get the complementation assertion we have to modify the proof above. Now let I be
the set of triples (E, F, 6), where E is a finite-dimensional subspace of X**, F a finitedimensional
subspace of X*, and 6 > 0. Introduce a partial order on I by (El, F1,61) <
(E2, F2, 62) if El C E2, F1 C F2, and 61 > 62, and let/g be an ultrafilter on I which refines
the partial order filter. For every (E, F, 6) E I let TE,F,e :E ~ X be the operator given by
the principle of local reflexivity. Using these operators we define as above an isometry T
from X** into XU. Define a map S from Xu into X** by S({xi }) = w*-limu xi. From the
properties of {TE,F,~} one deduces easily that ST is the identity on X**, so that T S is a
projection of norm 1 from XU onto T X**.
A property (P) of Banach spaces is called a super property provided that if X satisfies
(P) and Y is finitely representable in X, then Y satisfies (P). In particular, a super property
passes from a space X to all closed subspaces of its ultraproducts. So if (P) is a hereditary
property (i.e., passes to closed subspaces), a Banach space X has super (P) if and only if
every ultrapower of X has (P). For example, X is superreflexive if every ultrapower of X is
reflexive. An explicit local property which characterizes superreflexivity is the following:
A Banach space X is superreflexive if and only if for every 6 > 0 there is an integer N(6) so
that any 6-separated dyadic tree in the unit ball of X has height ~< N (6). By an 6-separated
dyadic tree of height N we mean a set of points {Xi,n: 1

Basic concepts in the geometry of Banach spaces 57
in the unit ball of X. In a similar manner we get in every nonreflexive space a 1-separated
dyadic tree of an arbitrary finite height in the unit ball.
It is not hard to show that the existence of arbitrarily tall e-separated trees in the unit ball
(for some e > 0 or for e = 1) is a selfdual property and thus so is superreflexivity. Much
deeper is the fact that a space is superreflexive if and only if it has an equivalent uniformly
convex norm. Since uniform convexity is defined by an inequality involving four vectors
the if part is obvious. The hard part is the only if part; that is, the construction of a uniformly
convex equivalent norm in a space which does not have e-separated dyadic trees of large
height in its unit ball. The notion of dyadic trees reminds one of martingales and in fact the
most elegant way to prove the only if part is to use vector-valued martingales. The proof
shows in particular that if X is uniformly convex then there is an equivalent norm whose
modulus of convexity ~(~) satisfies ~(e) ~> C~ q for some C > 0 and q < oe. The proof of
this fact and related material can be found in [2, 4.IV] and [6, IV.4] (see also [29]).
We introduce next a class of Banach spaces defined in terms of their finite dimensional
subspaces which are closely related to Lp(#) spaces. Let 1 ~< p ~< oe and )~ ~> 1. A Banach
space X is called an 12p,z space if for every finite dimensional subspace E of X there is
/7
a further subspace F D E with d (F, gp) ~< )~ where n -- dim F. A space is called an s
space if it is an s space for some )~ < oo.
It is evident that every Lp(#) space, 1 ~< p ~< oe, is an s space for every e > 0
(take as F a small perturbation of a subspace spanned by a finite number of suitable disjoint
indicator functions). Similarly, every C(K) space is an E~,l+~ space for every e > 0 (use
partitions of unity). By the principle of local reflexivity, if X** is an s space then X
is an s space for every ~ > 0. From ultraproduct arguments used already above, it
follows that: Any s space is isomorphic to a subspace of Lp(lZ) for some measure #
(1 ~< p ~< ~). In particular, for 1 < p < oe every s space is reflexive and any/22 space
is isomorphic to a Hilbert space.
It is possible to make a stronger statement. Assume first that 1 < p < oe and that X
is an s space. There is thus a set I of finite dimensional subspaces E of X, directed
by inclusion, with X -- ~EcI E, so that for every E there are operators TE'E --> gp(E),
SE'g~ (E) ---> E so that IITEII ~< 1, IISEII ~< ~ and SETE -- Ie (n(E) --dimE). LetL/be
an ultrafilter on I which refines the order filter and put Y -- (FI gp(E))z4- Define T from
X to Y by mapping x to the class represented by {TEx}Ecl (in view of the choice of/2
it does not matter that TEx is defined only for E which contain x). Define S: Y -+ X by
S{yE} = w-lim~ SEyE (recall that X is reflexive). Then ST = Ix and Y is an Lp(#) space.
Consequently: Every E,p space X, 1 < p < oo, is isomorphic to a complemented subspace
ofan Lp(#) space. If X is separable we deduce that X is isomorphic to a complemented
subspace of L p (0, 1). Similar considerations (starting with the fact proved in Section 4 that
g2 is isometric to a complemented subspace of Lp(O, 1)) yield that every Hilbert space is
isometric to a complemented subspace of some L p (#) space when 1 < p < oe.
In the cases p -- 1 and p -- cx~ we can reason similarly, but now define S'Y --+ X**
by S{yE} -- w*-limu SEyE and get that ST -- Jx, the natural inclusion of X into X**.
By considering T**" X** ----> Y** and S**" Y** ----> X (iv) and recalling that there is a norm
one projection from X (iv) onto X** (see Section 2), it follows that if X is an 121 space
then X** is isomorphic to a complemented subspace of an L l(#) space. Since L1 (#)* is

58 W.B. Johnson and J. Lindenstrauss
injective and since there is a norm one projection from X*** onto X* we deduce that if X
is an E1 space then X* is an injective Banach space. Similarly, if X is an s space then
the constructed ultraproduct Y is a C(K) space and X** is isomorphic to a complemented
subspace of the injective space Y**. That is, if X is an 12~ space then X** is an injective
Banach space.
The converse of the previous statements essentially hold. First we show: Let 1 < p < ~.
A Banach space X is isomorphic to a complemented subspace of an Lp(#) space if and
only if X is an s space or X is isomorphic to a Hilbert space. To see the "only if"
direction, assume that there is a projection Q from Y = Lp(#) onto a subspace X which is
not isomorphic to a Hilbert space. By the dichotomy principle for L p spaces, 2 < p < cx~,
discussed in Section 4 (and, by duality, also for 1 < p < 2), X has a subspace Z isomorphic
to e p onto which there is a projection, say R. Let E be a finite dimensional subspace of X
and ~ > 0. There is a finite dimensional subspace F of Y (a small perturbation of the span
of disjoint indicator functions) containing E so that d(F, ep)

Basic concepts in the geometry of Banach spaces 59
the structure of their finite dimensional subspaces and by passing to the duals we get direct
information only on the finite dimensional quotient spaces.
The s spaces give a nice local description of the complemented subspaces of Lp(O, 1)
for 1 < p < cx~. From the global point of view these complemented subspaces are very hard
to describe. Some global structure theorems are known (e.g., every separable s space
1 ~< p ~< ~ has a Schauder basis, and every separable infinite dimensional s space 1 <
p < cx~ which does not contain a copy of e2 is isomorphic to e p) but many natural questions
on them are still open. For example, it is unknown whether every separable s space,
1 < p < cx~ (p :/: 2), has an unconditional basis. There are uncountably many distinct
isomorphism types among the separable s spaces, 1 ~< p ~< ~ (p r 2). The simplest
examples for 1 < p 7~ 2 < oc (besides Lp(#) spaces) are ep | e2 and (e2 (~ e2 (~'" ")p.
A discussion of all of these questions is contained in [22].
Among the separable/2~ spaces there are the C(K) spaces with K countable compact
metric which, as was pointed out in Section 4, form uncountably many distinct isomorphism
types. There are however spaces X such that X* is isometric to el but X is not
isomorphic to a C(K) space. Such a space X is an s space for every e > 0. In fact,
a space X is s l+~ for every E > 0 if and only if X* is isometric to LI(#) for some
measure #. It is known that such an X has a subspace isometric to co. In contradistinction
to this, there are s spaces with 2. > 1 which have the RNP (and thus in particular do
not contain a subspace isomorphic to c0). For a discussion of s spaces see [40].
There are a continuum of isomorphism classes of separable s spaces. The simplest
example (besides el and L 1(0, 1)) is the kernel of a quotient map T :el --+ L1. The isomorphism
type of this kernel turns out not to depend on the choice of T. This kernel is an
s space which is not isomorphic to a complemented subspace of an L l(#) space. For a
discussion of s spaces, see [22].
There is also a "local view" of Banach lattices. A Banach space X is said to have
Gordon-Lewis local unconditional structure (GL-l.u.st.) if there is a constant 2. so that
for every finite dimensional subspace E of X there is a space Y with an unconditional basis
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, then we saw in Section 5 that for any finite
dimensional subspace E and every ~ > 0 there are vectors {xl ..... Xm } in X whose span
contains E and which is an e-perturbation of a sequence of disjointly supported vectors.
Since a disjointly supported sequence in a lattice is 1-unconditional and the bidual of a
Banach lattice is an order complete lattice, the general case follows from the principle of
small perturbations and the principle of local reflexivity.
It is clear from the definition that s spaces, 1 ~< p ~< ec, have 1.u.st.
The close relationship between 1.u.st. and lattice structure is expressed in the following
result: A Banach space X has GL-l.u.st. if and only if X** is isomorphic to a complemented

60 W.B. Johnson and J. Lindenstrauss
subspace ofa Banach lattice. Indeed, since by the definition a complemented subspace of
a space with GL-l.u.st. has GL-l.u.st., it follows that if X** is a complemented subspace
of a lattice it has GL-l.u.st. By the principle of local reflexivity we deduce that also X has
GL-l.u.st.
To prove the other direction we use ultraproducts. Assume X has GL-l.u.st. with some
constant )~. For every finite dimensional subspace E of X there are a space YE with 1-
unconditional basis and operators Tt; :E --+ Yt; and St; :Y t; --+ X with l[ Tt; 11 ~< 1, liSt; 11

Basic concepts in the geometry of Banach spaces 61
whether the disk algebra has the UAE As a matter of fact, the only general class of spaces
other than L p spaces which are known to have the UAP are the reflexive Orlicz spaces9
10. Some special classes of operators
In this section we shall discuss some special classes of linear operators between Banach
spaces and their relation to the geometry of Banach spaces. We shall also discuss some
other topics in operator theory which are relevant to the study of the structure of Banach
spaces.
Most of the classes of operators we consider have the ideal property, meaning that the
operators from a fixed X into a fixed Y having this property form a linear space and that
whenever T : X --+ Y belongs to this class then for every bounded U : Z --+ X and V : Y --+
W the operator V T U belongs to this class. The three most elementary operator ideals are
the bounded operators, the compact operators, and the weakly compact operators. Since
B(X, Y), K(X, Y), and WK(X, Y) are all Banach spaces under the operator norm, the
operator norm is the natural norm to use for operators in these classes. A Banach space
X has the Dunford-Pettis (DP) property provided that every weakly compact operator
with domain X maps weakly compact sets into norm compact sets. It is clear that if T E
WK(X, Y), S E WK(Y, Z), and Y has the DP property, then ST is a compact operator.
In particular, if X has the DP property and P is a weakly compact projection on X, then
p = p2 is compact. This means that the only complemented reflexive subspaces of X are
the finite dimensional ones.
A subset of a Banach space is relatively weakly compact if and only if it is relatively
weakly sequentially compact. Therefore, a space X has the DP property if and only if every
weakly compact operator with domain X maps sequences which converge weakly to zero
into sequences which converge in norm to zero.
The following is an elegant characterization of spaces having the DP property. X has
the DPproperty if and only if whenever {Xn}n~__l in X tends weakly to 0 and {x*}n~=l in X*
tends weakly to 0 the sequence of scalars {x* (Xn) }n~__l tends to O. Indeed, assume that X has
~< 11)
DP and x n > 0 9 The operator T" X ~ co defined by Tx -- (x* 1 (x), x 2 *(x) .... ) is a weakly
compact operator (it is easier to check that T* is weakly compact). Hence, if Xn --+ 0
weakly, Ix*(xn)[ ~< IITxnl[ ~ 0. Conversely, assume the condition on the sequences is
satisfied, T:X --+ Y is weakly compact, and x~ w> 0. If II Txn IIr 0 we may assume, by
passing to a subsequence, that II Txn II >~ 6 for some 6 > 0 and all n. Let y* E X* be such
that y*(Txn) -- IlTxn[I and IlYn* l[ - 1 for all n. Since T* is weakly compact as well we may
assume (passing again to a subsequence if needed) that T* y*
0- lim(T*y* - x*)(Xn) --limy*(Txn) --lim IITxn II
n /7 n
to
> x* for some x*. Then
a contradiction.
It follows from the criterion we just proved that X has the DP property if X* has this
property. (The converse is false: ( Y~n=l ~)1 has the DP property but ( Y~m= er 1 e2) n oo contains
a complemented copy of ~2 and hence fails the DP property.) It is also clear that a

62 W.B. Johnson and J. Lindenstrauss
complemented subspace of a space with the DP property has the same property. We shall
verify shortly that C(K) spaces have the DP property. Consequently we get that: All E~
and s spaces have the DP property.
To show that C(K) has the DP property let {xn},~__l be a sequence in the unit ball of
C(K) so that x,(t) --+ 0 for every t 6 K. Let {lZn},e~=l be a sequence in the unit ball of
C(K)* which tends weakly to 0. All the #, can be considered as elements in L1 (#) where
-- ZnCX~=l I#, 1/2 n. By the analysis done in Section 4 of sets in Ll (#) which have weakly
compact closure it follows that for every E > 0 there is a & > 0 so that whenever E is a
Borel set in K with #(E) < 6 then I#nl(E) < ~ for all n. By Egoroff's theorem there is
a set E with #(E) < & so that IXn(t)[

Basic concepts in the geometry of Banach spaces 63
It follows from this observation that the sum of two strictly singular operators is strictly
singular and thus the class of strictly singular operators has the ideal property. It is also
easy to check that SS(X, Y) is complete in the operator norm.
Two Banach spaces X and Y are called totally incomparable if there is no infinite dimensional
space Z which is isomorphic to subspaces of both X and Y. If every operator
from X to Y is strictly singular (and in particular if X and Y are totally incomparable)and
Z is a complemented subspace of X • Y then there is an automorphism T of X 9 Y so that
T Z is of the form T Z = Xo 9 Yo where Xo (respectively Yo) is a complemented subspace
of X (respectively Y). This result is not as simple as the preceding results in this section.
Its proof can be found in [ 11, 2.c. 13].
Another class of operators is that of Fredholm operators. An operator T:X --+ Y is
called a Fredholm operator if or(T) := dimker T < c~ and TX is closed with/~(T) :=
dimY/TX < cx~. The number i(T) = or(T) -/3(T) is called the index of T and has a
significant role in many applications. As a matter of fact i (T) can be defined also for non-
Fredholm operators provided TX is closed and at least one of the numbers or(T) or/3(T)
is finite (in this case the index can be of course either +cxz or -cx~). The class of Fredholm
operators is not closed under addition or composition. However if T1 and T2 are Fredholm
operators so is TiT2 (provided it is properly defined) and i(T1T2) = i(T1) + i(T2). Also
i(T*) = -i (T). These facts are simple exercises in linear algebra. Somewhat harder to
prove is that Fr(X, Y) is preserved by strictly singular perturbations. If T, S : X --+ Y with
T Fredholm and S strictly singular then T + S is also Fredholm with i(T + S) -- i(T) (see
[11, 2.c.9]).
An operator T'X --+ Y is called absolutely summing if whenever EiC~=l xi converges
unconditionally in X the series Ei~l Txi converges absolutely. From the closed graph
theorem one deduces easily that T is absolutely summing if and only if there is a constant
C so that for any choice of {xi }in=l in X
EII Txi ]l ~ C sup I x*(xi)]" I[X* II ~ 1 .
i---1
i--1
More generally, for every 0 < p < cx~ we can define the class of p-summing operators as
those operators for which there is a constant C so that for all choices of {xi n }i=1 in X
IlTxill p

- sup
64 W.B. Johnson and J. Lindenstrauss
Without further mention we shall henceforth assume that p ~> 1 when discussing p-
summing operators. However, there are important applications to Banach space theory of
p-summing operators for 0 < p < 1 (see, e.g., the article [32]), and much of the theory of
p-summing operators, 1 ~< p < oo, goes over to the range 0 < p < 1. In particular, there
is a version of the Pietsch factorization theorem (discussed below when 1 ~< p < oo) for
p-summing operators, 0 < p < 1 (see [20, Theorem 9.1]).
Notice that the supremum on the right side of (28) is the norm of the operator from s
to X defined by ek ~ xk (notice that p ~> 1 is needed for this). Thus top(T) can also be
defined by
Zcp(T) -
[[TVenl] p " V " g.p, --~ X, []VI] ~< 1 . (29)
Equation (29) implies that 7rp(T) is the supremum of rcp(TV) as V varies over the norm
one operators from s into X.
In contrast to the classes of weakly compact or strictly singular operators, the class of p-
summing operators is determined by the behavior of T on the finite dimensional subspaces
of X. Because of the quantitative nature of its definition the notion of p-summing norm
plays also an important role in the study of finite dimensional spaces.
If K is a compact Hausdorff space and/z a regular probability measure on K then the
identity operator Ip from C(K) into Lp(#) is easily seen to be a p-summing operator
with p-summing norm one. It turns out that this simple example is the prototype of general
p-summing operators. This is the content of the Pietsch factorization theorem. Let X be a
subspace of C (K). An operator T : X --+ Y is p-summing, 1 Xp
Y

Basic concepts in the geometry of Banach spaces 65
(For p -- 2, the operator S can be extended to an operator from L p (#) into Y, but for p :/: 2
this is not in general the case.)
To prove the Pietsch factorization theorem, assume that 7rp(T) = 1. Consider the following
subsets of C(K)
G = {f e C(K): sup{f(t)" t 6 K} < 1},
F = conv{f E C(K): f(-)- Ix(.)l z', IITxll- 1}.
The sets F and G are convex with G open and, since 7rp(T) = 1, F M G = 0. By the
separation theorem there is a regular signed measure # on K and a positive )~ so that
ffd#)~ for f EF. Since wheneverg~< f and fEG
also g E G it follows that the measure/z has to be positive so after normalization we may
assume that it is a probability measure. Also since G contains the open unit ball of C(K)
we must have )~ >~ 1. Hence for every x ~ X, IITx[I p /.... If the sequence is infinite, then
necessarily )~n(T) --+ O. For the Banach space ~2, the spectral theorem for compact op-

66 W.B. Johnson and J. Lindenstrauss
erators says that if T ~ K(H, H), then there is an orthonormal basis {Xn}nC~ 1 SO that the
self-adjoint operator T*T is represented as
T*Tx -- Z )~n(T*T)(x,xn)Xn (31)
nEP
with )~n(T*T) > 0 for n 6 P and T*Txn- 0 for n r P.
Now if X is a Banach space with a 1-symmetric basis S- {en}n~__l , let S(62) be those
compact operators on 62 for which ~ ~/)~n(T*T) en converges in X, and, for T 6 S(62), set
Then (S(62), o's) is a Banach space (verifying the triangle inequality is a bit tricky). It is
less difficult to check that S(62) satisfies the unitary ideal property
crs(UTV)

Basic concepts in the geometry of Banach spaces 67
orthonormal basis {en}n~__| and )~, ~> 0 so that Ten -- )~nen for every n and 1 -- (re(T) --
~nC~=l)~n .2 Let {~n}~_|_ be a Rademacher sequence. Using Khintchine's inequality (1) we
get
sup I(xi,x)[" Ilxll ~< 1 , ~> sup xi, Z-l-)~jej
i--1 -+- i--1 j--1
>>. ~ E ~_~)~j~j (xi, e~) >1 el ~_~ I,~j (xi, ej)l 2 -- A1 II Zxi II.
i=1 j--1 i=1 j=l i=1
This shows that 7rl (T) ~< A11 crz(T).
To show the converse, it is enough to check that if T 6//p(~2, ~2) with 2

68 W.B. Johnson and J. Lindenstrauss
norm at most one when considered as an operator from s to s Let H be a Hilbert space.
Then IIA @ In II ~< KG as an operator from s (H) to s (H), where
(A | IH)(yl ..... Yn) "= aljyj ..... anjyj
j=l
for (Yl ..... Yn) in s
To prove Grothendieck's inequality fix n and a matrix A- (aij)in, j=l with [IAll "-
IIAs ---> s 1 as above, and put !A! := [IA | IH's ---> s We have to
find an estimate on !A! independent of n and A. Note first that for all ui and vj in a Hilbert
space
j=l
aij (ui, v j)
i,j=l
~< !A! max Ilui II max Ilvj II.
i j
Since all Hilbert spaces are created equal, we can use for H any (infinite dimensional or
even 2n-dimensional) Hilbert space. For the purpose of proving Grothendieck's inequality,
some Hilbert spaces are more equal than others! We use for H the subspace of L2(0, 1)
mentioned in Section 4 consisting just of functions having a Gaussian distribution with
mean 0. To prove Grothendieck's inequality it is enough to consider norm one vectors
{Xi}in=l and {YJ}j=I in H. Given 0 < 6 < 1/2 there is an M -- M(6) so that for any norm
one function f in H, Ilf - f M 112 --" • where
fM (t) - ] f (t) if If(t)l ~< M,
M sign f(t) if If(t)[ > M.
[
Note that by our assumption on the matrix A, for any choice of functions f/ and g j in
L2 (0, 1) which are uniformly bounded by M,
Zaij(fi,gj) f01 Z aij fi(t)gj(t)
9 .
i,j t,j
dt ~< M 2.
Hence
Z aij (xi, yj )
i,j
Zai,j(XiM, y M}
i,j
+ Zaij(xi, Yj -- Y M)
i,j
+ Z aij(xi- x M, yM)
i,j

Basic concepts in the geometry of Banach spaces 69
inequality. With the optimal choice of 6 this proof yields that Kc < 8.69 (or KG < 8.55 if
one exercises more care in the computation), but it is known that KG < 1.79. Incidentally,
although the best value for Kc is unknown, it is known that for complex scalars it is smaller
than for real ones.
The first application of Grothendieck's inequality is: Every bounded linear operator
T from s to ~2 is absolutely summing and rrl(T) ~ KGIITII. Indeed, let {ej}7_ 1 be the
unit vector basis for s and let ui -- Y'~jm= 1 aijej be n vectors in s for some m so that
Ein=_ , m of scalars of
1]x (ui)l ~< 1 for every unit vector x* in s For any choice {Sj}j= 1
absolute value

70 W.B. Johnson and J. Lindenstrauss
.....
define in g~c ( H) the vector 2 "- (yl ym).Thenllxlle~(.)
__ ( n 1/2
II Y~k=l Ixkl 2) I1~ ~< 1.
Similarly,
k=l
ITxkl 2
-II T o
and this last quantity is at most KGIITII by (the conceptual form of) Grothendieck's
inequality. This gives (33) when X- ~, Y- g~n and hence also when X = C(K),
Y = L1 (#).
Suppose now that X and Y are general Banach lattices and T :X --+ Y has norm one.
Let x l ..... Xn be in X with II (~i=1 n Ixil 2) 1/2 II- 1 and set u -- (y~in__l Ixil2) 1/2 Since we
are interested only in estimating II(Y~i~I I Txi 12) 1/2ll, we can assume by replacing X by
the (separable) sublattice generated by x l ..... Xn that X is separable. The space T X is
then separable, so we can similarly assume that Y is separable. Given any ~ > 0, as seen in
Section 5 there is a strictly positive functional y* E Y* with
y* ITxil 2
i=1 i:,
1/2
and IlY*II ~ 1 -q- ~. As mentioned in Section 5, the space Xu is lattice isometric to a C(K)
space and Yy, is lattice isometric to an L l(#) space. The natural lattice homomorphisms
J1 : Xu --+ X, J2 : Y --+ Yy* satisfy II J1 II = Ilu II = 1 and II J2 II = IlY* II ~< 1 + ~. We can then
apply (33) to the operator J2 T J1 to obtain that
1/2
i=1
Irxil 2
Yv*

Basic concepts in the geometry of Banach spaces 71
infinite dimensional). Note that ~2 is isometric to a subspace of L1 (0, 1) (see Section 4)
and thus ~ -- ~2 is also a quotient space of C(0, 1) because ~2 is reflexive and C(0, 1),
considered as a subspace of L~ (0, 1) -- L 1(0, 1)*, determines the norm of L1 (0, 1). (If Y
is a reflexive subspace of a Banach space X and Z is a norm closed subspace of X* which
determines the norm of X, then the restriction operator z* ~ z~y is a quotient mapping
from Z onto Y*.)
We will next give a glimpse into the connections of p-summing operators to the theory
of finite dimensional spaces discussed in Section 8. If X is the space g~ then 7f2 (Ix) =
since this is the Hilbert-Schmidt norm of the identity in g~. It is quite surprising that: For
any X with dim X = n, re2 (Ix) = ~/-ff. To see this take any norm one operator V from ~2
into X. By projecting onto the orthogonal complement of the kernel of V, we can factor V
as V1P where VI"~ n -+ X, P's --+ ~n, II P II - 1, II V1 II - II V II, and m ~< n. Then
yr2(Ix g) ~ IIIxll IIV~ll~2(IeT)llPII ~ v/-n,
Taking the supremum over all such V gives Yr2 (Ix) ~ ~/-n.
To prove the other inequality, we get from the Pietsch factorization theorem a probability
measure # on Bx. and T :L2(#) --+ X so that TI21x = Ix and IITII = 7rz(Ix), where as
usual I2:C(Bx.) --+ L2(#) is the formal identity and X is canonically embedded into
C(Bx.). The space X2 := I2X is an n-dimensional Hilbert space and IzT is the identity
on X2. Hence
--- 7F2(Ix2) ~ 7r2(I2)llTII = 7r2(Ix).
As a consequence of the preceding we deduce: The projection constant ofan n-dimensional
space X is at most v/if; that is, whenever Y contains X there is a projection P from Y onto
X with I[ P [[ ~< v/-~. Indeed, since fez(Ix) = V/if, the Pietsch factorization theorem yields
that the identity operator on X can be represented as X J > Loo(/z) 12> L2(#) v > X,
with J an isometry, 12 the formal identity, and II vii- v/if, Since L~(#) is 1-injective
the operator J can be extended to a norm one operator T from Y into L~(#). The operator
P :-- V12 T is a projection from Y onto X with norm at most v/ft. Notice also that
the factorization used above gives another proof of the result proved in Section 8 that
d(X, ~) ~ x/~.
The estimate of x/~ for the projection constant of an n-dimensional space is essentially
sharp. This is discussed in [33].
The p-integral operators form a class of operators which are closely related to the p-
summing operators. An operator T :X --~ Y is said to be p-integral, 1 ~ p ~ cx) (in symbols
T E Zr(X, Y)), provided that the composition Jr T of T with the canonical embedding
Jy :Y --~ Y** factors through the formal identity I~,p : L~(#) ~ Ln(#) for some
probability measure #:
L~(#)
I~, p
> Lp(#)
AI T JY ~B
X > Y > Y**
(34)

72 W.B. Johnson and J. Lindenstrauss
The p-integral norm ip(T) is then defined to be the infimum over all such factorizations
of [[ A [[ [[ B [[. By taking ultraproducts one sees that this infimum is really a minimum. The
space (Zp(X, Y), ip) is easily seen to be a Banach space and ip satisfies the ideal property
ip(STU) ~ []S[[ip(T)[[U[[.
If T is in Zp(X, Y) and X is a subspace of C(K), with K a compact Hausdorff space,
then there is a probability measure v on K and an operator S:Lp(v) ---+ Y** with [[S[[ ----
ip(T) which makes the following diagram commute:
C(K)
Ip
> Lp(v)
X
T
> Y
Jr
> Y**
(35)
Indeed, if (34) holds, A can be extended to an operator A" C(K) --+ L~(#) because
L~(#) is 1-injective. By the Pietsch factorization theorem there is a probability measure
v on K so that for each x E C (K),
[[BI~,pAx[] ~ 7rp(Bl~,pA)
[x[ p dv
and the desired conclusion follows from
7rp(BIc~,pA) ~ IIBllTrp(l~,p)llS, II =
IIBlllIAII.
It is evident that zrp(T) ~ ip(T) and from the Pietsch factorization theorem it follows
that Zcp(T) = ip(T) if the domain of T is a C(K) space. As was mentioned implicitly in
the discussion of 2-summing operators, 7r2(T) -- i2(T) for any operator. One reason for
defining p-integral via a factorization of Jy T rather than T is that this forces i p(T) --
ip(T**) (use the fact that a dual space is norm one complemented in its bidual).
The 1-injectivity of C(K)** gives that a p-summing operator T into a C(K) space is
p-integral with ip(T) = 7rp(T). The equality il (T*) = il (T) follows from the observation
that the adjoint I~ of 11 "C(K) --+ Ll(v) is l~,l'L~(v) --+ Ll(v) followed by the identification
of L1 (v) with the norm one complemented subspace of C(K)* consisting of the
finite signed measures which are absolutely continuous with respect to v.
For other values of p, the adjoint of a p-integral operator need not be strictly singular
(see [9, 5.12]) and hence need not be q-summing for any q < c~.
For each 1 ~< p ~< cx~, p -r 2, there exist p-summing operators which are not p-integral
(see [9, 5.13]). The case p -- 1 is particularly easy to deduce from the theory we have presented.
We saw that every operator T :~1 --+ ~2 is 1-summing. If T :~1 --+/~2 is 1-integral,
A I~, 1 B
then T has a factorization ~1 > L~ (#) > L1 (/z) > ~2. But then B and also I~, 1 are
1-summing, hence B I~, l, whence also B I~, 1A = T, are compact (use the fact that L 1 (/z)
has the DP property).
For p = c~ and Y reflexive the infinity integral operators from X to Y are exactly those
which factor through some L~(#) space (with the integral norm equal to the best factorization).
In particular for X reflexive the infinity integral norm of the identity of X is

- 1/x
Basic concepts in the geometry of Banach spaces 73
finite if and only if X is finite dimensional (and is equal to the projection constant of X in
that case). Thus also for p -- cx~ it is evident that summing and integral norms can be very
different.
The main reason for introducing p-integral operators is that they are needed for the
duality theory of lip(X, Y). For simplicity, we restrict to the case where X and Y are
finite dimensional. Following the notation used in Section 8, for finite dimensional X,
Y and ot a norm on B(X, Y) we represent the dual of (B(X, Y), or) as (B(Y, X),ot*),
where the pairing is given by (S, T) = trace TS (= traceST). Then for all 1 ~< p ~< cx~,
Hp(X, Y)* = Zp,(Y, X) when X and Y are finite dimensional. This just means that for
each S E B(Y,X), ip,(S) = sup{traceTS: T E B(X,Y),rcp(T) L1 (#)
A'I S ~B,
Y >X
(37)
one has for E > 0 an operator A~ :Y --~ L~(#) with [[A1 - Aell < ~ so that A~Y is contained
in the simple functions. This gives a factorization of B11~,l A~ of the form (36)
with IIBIIIIAIIIIAII ~< IIB~ II III~,~ IIIIA~ II. Setting N := dim Y, we get A[(S - BII~,IAE)

74 W.B. Johnson and J. Lindenstrauss
and
The operator S** B Ip satisfies
ir(S**BIp) -- 7cr(S**BIp)

Basic concepts in the geometry of Banach spaces 75
space Z can be replaced with the algebraic sum Z(X) "-- Xo + X1 C Z topologized with
the (complete) norm
Ilxll - inf[ Ilxollto] -+- Ilxl Ill1]" x - xo -4- Xl, xo E Xo, xl ~ xl },
where I1" I1[i1 is the norm of Xi. We always take for Z the space Z(X). The space A(X) "--
Xo N X1 also plays a role in the theory; it is naturally normed by the (complete) norm
IIx II - IIx Ilto] v IIx lit 1].
Thus we have for i -- 0, 1 inclusions
Z(X)
with Ilni II ~ 1, IIJi II ~ 1. Any Banach space X satisfying A(X) C X C Z(X) with both
inclusions continuous is called an intermediate space between Xo and X1 (or an intermediate
space with respect to X). The spaces Xo, X1, A(X), and Z(X) are all intermediate
spaces with respect to X.
Given Banach couples X -- (Xo, X1), Y -- (Yo, Y1 ), and a linear mapping T" Z (X) ---->
Z(Y), we write T E B(X, Y) provided Tix i E B(Xi, Yi) for i --0, 1.
If X and Y are intermediate spaces with respect to X and Y, respectively, we say that
X and Y are an interpolation pair for X and Y provided that if T E B(X, Y), then TIx
B(X, Y). If always
II T II x, Y ~ II T II go, Yo v II T II x,, Y,
X and Y are said to be an exact interpolation pair for X and Y, where IlSlJx, Y :--
][Slxl]B(X,y ) (if SX r Y then I]Sllx, Y "- oc). Finally, if 0 ~< 0 ~< 1 is such that the inequality
1-0 0
II T IIx, Y ~ cII TIIxo,Yo. II T IIx,,y,
always holds, then the interpolation pair X and Y are said to be of exponent 0 (and exact
of exponent 0 if C -- 1). When X -- Y and X -- Y, we abbreviate by saying, e.g., that X is
an interpolation space with respect to X provided X and X are an interpolation pair with
respect to X and X.
The most classical realization of this abstract set-up occurs in the scale of Lp(#) spaces.
For 1 ~< p0, pl ~< oc, the pair (Lpo(#), Lpl (#)) is a Banach couple, and for p0/x pl ~<
p ~< P0 v pl the space Lp(l z) is an intermediate space between Lpo (/z) and Lpj (#). In the
language of interpolation theory the Riesz-Thorin interpolation theorem can be stated as
follows. Let 1

I
76 W.B. Johnson and J. Lindenstrauss
Then for all measures tx and v, (complex) Lp(Ix) and Lq(v) are an exact interpolation
pair of exponent 0 with respect to (Lpo(ix), Lpl (Ix)) and (Lqo(V), Lq, (v)). It is formal to
derive from the Riesz-Thorin theorem that its statement remains true in the setting of real
scalars as long as P0 ~< pl and q0 ~< ql (or, what is the same, if pl ~ p0 and ql

Basic concepts in the geometry of Banach spaces 77
m
0 with respect to X and Y. To see this, let T 9 B(X, Y), x 9 X[ol, e > 0, and choose
f 9 7-/(X) so that f(O) -- x with IlfllT-t(~) ~< Ilxll[0] + e. Define g'S --+ S,(Y) by
g(z)-- IlZll%oly ollrll-=
X1, Y1 Tf(z)
-- 0-1 -0
Then g is in 7-/(Y), IlgllTt(F) ~< II f llT-t(N), and g(O) -IlTllxo,vollTll x,, Y, Tx so that
1-0
IITxll[ol ~ IITIIxo,YollTIl~
Ilgll~(y)
1-0

78 W.B. Johnson and J. Lindenstrauss
(Lpo (/Z), Lpl (/z))[0] --- Lp(/Z). The argument does use the following general fact about the
complex method. For any Banach couple X and 0 < 0 < 1, A(X) is dense in X[o] (see [4,
4.2.2]).
The identification Lp - (Lpo, Lpl)[O] , ~ 1 = -~-~-- 1-0 _ z- ~-~-, 0 has a generalization to Banach
lattices. Assume that X0 and X1 are complex Banach lattices of/z-measurable functions
which are ideals in the space of all/z-measurable functions, 0 < 0 < 1, and set
1 -
S 0 {(sign xoxl)lxoll-~
. .
~ xo ~ S0, Xl E Xl}
0 XOl ._
If Xo is order continuous then (Xo, Xl)[O] =
X~ -0 XO1 and
Ilxll[0] -inf{llx011[0] 1-0 Ilxl II~l] 9 Ixl-Ix011-~ Ix1 I ~ }. (38)
For a proof see Section iv.l.11 in [13]. The inequality

Basic concepts in the geometry of Banach spaces 79
Earlier we saw how the complex method can be used to construct the p-convexification
of a Banach lattice. The /C-method can also be used to construct Banach spaces with
interesting properties. Suppose, for example, that both X0 and X t have a normalized
1-unconditional basis, each of which we identify with the unit vectors {en}n~=l so that
X0 and X1 are both contained in co and thus X - (X0, X1) is a Banach couple. Let E
be another space with a normalized 1-unconditional basis, also denoted by {en}n~ Since
/C(X, E, a, b) is an exact interpolation space with respect to X, it follows that {en}n~__l is a
1-unconditional basis for/C(X, E, a, b) which moreover is even 1-symmetric if {en}~__l is
1-symmetric in both X0 and in X1 (one need only check that the span of {en }n~ is dense in
/C(X, E, a, b)). Assume now that {en}ncC__l is 1-symmetric in both X0 and X1, X0 C X1, and
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
and the weight sequence b satisfies bn 1' ec sufficiently quickly, then there are disjoint
finite subsets Cn of N so that the mapping en ~ II lc~ IlK;(N,a,b)lcn extends to an isomor-
phism from E onto a complemented subspace of/C(X, E, a, b). This is how one proves
the result mentioned in Section 3 that a space with an unconditional basis is isomorphic to
a complemented subspace of a space which has a symmetric basis.
"Good" properties (such as reflexivity and superreflexivity) possessed by the Banach
spaces X0 and X1 (often possessed by just one of the spaces) generally pass to interpolation
spaces between X0 and X l which are obtained by the complex method or by the E-method
(at least when the weight sequences a, b satisfy some growth conditions and the space E
is "nice"). Sometimes interpolation spaces even have a good property which neither X0
nor X1 possesses. For example, suppose that X0 C X1 (this can be relaxed but is good
enough for applications). If E is reflexive, the inclusion J" Xo ~ X1 is weakly compact,
O0
}~,n=l an < oc, and bn "~ oc, then/C(X, E, a, b) is reflexive. For a proof when E- ~2,
which is easily modified to cover the general case, see [ 15, 2.g. 11 ]. A consequence of this
result is the following factorization theorem for weakly compact operators. If T" X ~ X1,
is weakly compact then T factors through a reflexive space; that is, there exists a reflexive
Banach space Y and operators A ~ B(X, Y), B ~ B(Y, X1) so that T -- BA. To derive
this factorization theorem from the interpolation result mentioned above, it suffices to take
Y-/C(X0 X1 ~2 { 2-n }n= ec 1 { 2n } oc 1) where X0 is the span in X of T Bx with T Bx as
the unit ball of X0. Let A be T, considered as an operator into Y, and let B be the formal
inclusion from Y into X l. Then A and B are operators and T -- BA.
We turn now to the realization of the /C-method which is used most often in analysis
and is discussed extensively in the books on interpolation theory we have mentioned.
Given a Banach couple X- (Xo, X1), 0 < 0 < 1, and 1 ~< p < ec, Xo,p denotes the space
/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-
stead of using [I 9 II/c(N,a,b~ on Xo,p, it is customary to use
Ilxll0,p --inf [le~ v II e- -~
where the infimum is over all x0(t), Xl (t) for which e ~ xo(t) E Lp(R, X0), e-(1-~ (t) c
Lp(R, X1), and x - xo(t) + xl (t) for every t in R. The expression II 9 I]0,p is a norm on
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. Lindenstrauss
is uniformly convex (rather than just superreflexive) if either X0 or X 1 is uniformly convex
and 1 < p < cx~ [15, 2.g.21]. Moreover, I]" IIO,p is very good for interpolation purposes,
for if Y is another Banach couple then (Xo,p, ]]. ]]0,p) and (Yo,p, ][" ]]0,p) are an exact
interpolation pair of exponent 0 with respect to X and Y.
It is of course important to identify Xo,p when X is a concrete Banach couple. The
spaces that arise in this connection when X is a couple of L p(#) spaces are the L p,q
spaces. Given 0 < p < ~, 0 < q < cx~, a measure #, and a Banach space X, Lp,q (#, X)
is the space of X valued strongly measurable functions x for which
I]Xllpq "-- q/p [tl/px.(t)]q dt 1/q < o~, (42)
( f0 t )
where x*(t) is the decreasing rearrangement of IIx(t)llx. For 0 < p ~< cx~, Lp,~(#, X) is
the space of X valued strongly measurable functions for which
IlXllp,~ "-suptl/Px*(t) < ~. (43)
t>0
When X is the scalar field we write Lp,q(#). Evidently Lp,p(#, X) = Lp(#, X). Note
that for p > q ~> 1, the space Lp,q(#) is the Lorentz function space Lw, q(#) defined
in Section 5 with W(t) --qt q/p-I and hence is a Banach space. When q > p ~> 1 the
expression II 9 I] p,q does not satisfy the triangle inequality, although II 9 Ilp,q is equivalent to
a norm when p > 1. In any case Lp,q (#, X) is a metrizable topological vector space.
The main result about spaces obtained from L p and L p,q spaces via the/C-method is
the following (see [4, 5.3.1]): Let 0 < Po, Pl, qo, ql 0, C-1]] 9 ]]p,q ~ ]] 9 ][O,q
C II 9 II p,q. When the expressions II 9 II Pi ,qi' i -- 0, 1, are equivalent to norms one can deduce
from (44) and earlier comments an interpolation theorem for operators. In fact, there are
cases where interpolation is valid even when the spaces are not all Banach spaces. In particular,
assume that T" Lpi,r i (l~, X) -+ Lqi,s i (19, X) is continuous for i -- 0, 1, with P0 ~: Pl,
q0 ~ ql, 0 < 0 < 1, and define p, q by 1/p - (1 -O)/po + O/pl, 1/q - (1 -O)/qo --
O/ql. If p ~< q, then T'Lp(lZ, X) ~ Lq(19, Y) is continuous and for 0

Basic concepts in the geometry of Banach spaces 81
that X is obtainable from X via the/C-method. This theorem says the following. If X is ~Cmonotone
for X, then there are E, a, and b so that X =/C(X, E, a, b), up to an equivalent
renorming.
The pair (Lp(O, 1), Lq(O, 1)), 1 ~< p < q ~< ~, is a Calder6n couple, and many other
examples are known (see [31]). One interesting problem which is not completely solved
is to determine a necessary and sufficient condition for a Banach couple of symmetric lattice
ideals on [0, 1] (see Section 5) to be a Calder6n couple. Much is known about the
interpolation spaces for such a couple. For example, an exact interpolation space with
respect to such a couple must itself be a symmetric lattice ideal (the first step is to observe
that if r'[0, 1] --+ [0, 1] is a measure preserving automorphism, then x w-> x(r)
defines an isometric automorphism of any symmetric lattice on [0, 1]). When the couple
is (L1 (0, 1), L~(0, 1)), it is reasonable to guess that the converse is true. It is not,
but: If L~(O, 1) is dense in the symmetric lattice ideal X, then X is an exact interpolation
space with respect to (Ll (0, 1), L~(0, 1)) [13, Theorem 4.10]. Most natural symmetric
lattice ideals on [0, 1], including all the separable ones, satisfy the hypothesis of this
theorem.
12. List of symbols
Here is a list of symbols used in this introductory article and, where appropriate, a reference
to where they are defined.
N
R
C
T
IP
E
Ap, Bp
KG
C(K;X)
C(K)
L p (l,t, X)
Lp(#)
L~(#,X)
Lp(O, 1)
Cp(~r)
ep(r)
The natural numbers.
The real numbers.
The complex numbers.
The complement of the set S.
The unit circle in the complex plane.
A probability measure (Section 2).
f. dip (Section 2).
The constants in Khintchine's inequality (1); also the constants in the
Kahane-Khintchine inequality (24).
The constant in Grothendieck's inequality (Section 10).
Continuous functions f on the (usually) compact Hausdorff space K
taking values in the (usually) normed space X, normed by ]lfll =
suPtcK II f(t) II.
C (K; X) when X is the scalar field.
The #-measurable X-valued functions f for which
Ilfllp := (f IIf[I F d#) 1/p < ~ (Section 7). Here 0 < p < ec.
L p (#, X) when X = R.
The #-measurable essentially bounded X-valued functions, with norm
11 f II ~ := inf~A=O sup I fl A I"
L p (#) when # is Lebesgue measure on the unit interval.
Lp(#) when # is normalized Lebesgue measure on the unit circle.
Lp(#) when g is counting measure on the set F.

82 W.B. Johnson and J. Lindenstrauss
~.p
17 ~p
C
co(F)
CO
L p,q (/z, X)
L p, oc (/z , X)
L p,q (/z )
Bx
Bx(x,r)
s-L
S_L
d(X, Y)
X ,~ Y
~x(.)
px(.)
Xu
Xu,
Ix
Jx
Ip
Ip,q
B(X,Y)
K(X, Y)
WK(X, Y)
SS(X, Y)
Fr(X, Y)
.N'(X, Y)
A/'(T)
M(P)(T)
M(p)(T)
Tp(X)
Cp(X)
g~p(F) when F = N.
g~p(F) when F -- { 1, 2 ..... n}.
The subspace of s of scalar sequences which have a limit.
The closure in s (F) of the scalar sequences which have finite support.
co (F) when F -- N.
The /z-measurable X-valued functions f for which ][f][p,q "=
(q fo ][tl/pf *(t)][q ~)l/q < co(Section 11). Here 0 < p,q < co and
f* is the decreasing rearrangement of []fllx.
The /z-measurable X-valued functions f for which [1 f ]]p,oc "=
supt>otl/Pl[f*(t)l[x < co (Section 11). Here 0 < p ~< co and f* is
the decreasing rearrangement of 11 f ]Ix.
Lp,q (/z, X) when X -- ]R.
The closed unit ball of the Banach space X.
The closed ball of radius r with center x in the Banach space X; denoted
also B(x, r) when X is understood.
All linear functionals which vanish on S (when S is a subset of a Banach
space).
The intersection of the kernels of all linear functionals in S (when S is
a subset of the dual of a Banach space).
The Banach-Mazur distance from X to Y (Section 2).
The space X is isomorphic to the space Y.
The modulus of convexity of the space X (Section 6).
The modulus of smoothness of the space X (Section 6).
When X is a lattice and u ~> 0, the abstract M-space which has the order
interval [-u, u] as the unit ball (Section 5).
When X is a lattice and u* ~> 0 in X*, the abstract L 1-space which is
the completion of X under the seminorm [[x ]]u. -- u* ([x[) (Section 5).
A Rademacher sequence (Section 4).
The identity operator on the space X.
The canonical embedding of X into X**.
The formal identity operator from C(K) to Lp(/z) (when/z is a finite
measure on the compact Hausdorff space K).
The formal identity mapping from Lp(/z) to Lq(/z).
The bounded operators from X to Y.
The compact operators from X to Y.
The weakly compact operators from X to Y.
The strictly singular operators from X to Y (Section 10).
The Fredholm operators from X to Y (Section 10).
The nuclear operators from X to Y (Section 8).
The nuclear norm of the operator T (Section 8).
The p-convexity constant of the operator T (Section 5).
The p-concavity constant of the operator T (Section 5).
The type p constant of the Banach space X (Section 8).
The cotype p constant of the Banach space X (Section 8).

Basic concepts in the geometry of Banach spaces
83
ITp(X, Y)
7rp(T)
In(X, Y)
ip(T)
The p-summing operators from X to Y (Section 10).
The p-summing norm of the operator T (Section 10).
The p-integral operators from X to Y (Section 10).
The p-integral norm of the operator T (Section 10).
References
[1] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York, 1985.
[2] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, Mathematics Studies 68, North-Holland,
Amsterdam (1985).
[3] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Coll. Pub. 48, Amer. Math.
Soc., Providence, RI (2000).
[4] J. Bergh and J. Lrfstrrm, Interpolation Spaces: An Introduction, Grundlehren der Mathematischen Wissenschaften
223, Springer-Verlag, New York (1976).
[5] Yu.A. Brudnyi and N.Ya. Kruglyak, Interpolation Functors and Interpolation Spaces I, North-Holland,
Amsterdam (1991).
[6] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Longman Scientific
& Technical, Essex (1993).
[7] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York (1984).
[8] J. Diestel and J.J. Uhl, Vector Measures, Mathematical Surveys 15, Amer. Math. Soc., Providence, RI
(1977).
[9] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced
Math. 43, Cambridge University Press, Cambridge (1995).
[10] R. Durrett, Probability: Theory and Examples, Duxbury Press (1996).
[ 11 ] P. Habala, P. H~ijek and V. Zizler, Introduction to Banach Spaces I, Matfyzpress, Univerzity Karlovy (1996).
[ 12] P. Habala, P. Hfijek and V. Zizler, Introduction to Banach Spaces II, Matfyzpress, Univerzity Karlovy (1996).
[13] S.G. Krein, Yu.I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Transl. Math. Monogr. 54,
Amer. Math. Soc., Providence, RI (1982).
[14] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Ergebnisse der Mathematik
und ihrer Grenzgebiete 92, Springer-Verlag, Berlin (1977).
[15] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II: Function Spaces, Ergebnisse der Mathematik
und ihrer Grenzgebiete 97, Springer-Verlag, Berlin (1979).
[16] V.D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes
in Math. 1200, Springer-Verlag, Berlin (1986).
[17] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94,
Cambridge University Press, Cambridge (1989).
[18] H.L. Royden, Real Analysis, 3rd edn., Macmillan, New York (1988).
[19] W. Rudin, Functional Analysis, 2nd edn., McGraw-Hill, New York (1991).
[20] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman
Scientific & Technical, Essex (1989).
[21 ] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics 25, Cambridge
University Press, Cambridge (1991).
Handbook articles which are referenced in this introductory article
[22] D. Alspach and E. Odell, L p spaces, This Handbook.
[23] D.L. Burkholder, Martingales and singular integrals in Banach spaces, This Handbook.
[24] EG. Casazza, Approximation properties, This Handbook.
[25] R. Deville and N. Ghoussoub, Perturbed minimization principles and applications, This Handbook.
[26] V. Fonf, J. Lindenstrauss and R.R. Phelps, Infinite dimensional convexity, This Handbook.

84 W.B. Johnson and J. Lindenstrauss
[27]
[281
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[381
[39]
[401
[41]
[421
[43]
[441
T. Gamelin and S.V. Kislyakov, Uniform algebras as Banach spaces, This Handbook.
A.A. Giannopoulos and V. Milman, Euclidean structure in finite dimensional normed spaces, This Handbook.
G. Godefroy, Renormings of Banach spaces, This Handbook.
W.T. Gowers, Ramsey theory methods in Banach spaces, This Handbook.
N.J. Kalton and S.J. Montgomery-Smith, Interpolation andfactorization theorems, This Handbook.
S.V. Kislyakov, Banach spaces and classical harmonic analysis, This Handbook.
A. Koldobsky and H. K6nig, Aspects of the isometric theory of Banach spaces, This Handbook.
M. Ledoux and J. Zinn, Probabilistic limit theorems in the setting of Banach spaces, This Handbook.
J. Lindenstrauss, Characterizations of Hilbert space, This Handbook.
E Mankiewicz and N. Tomczak-Jaegermann, Quotients offinite-dimensional Banach spaces; Random phenomena,
This Handbook.
B. Maurey, Banach spaces with few operators, This Handbook.
B. Maurey, Type, cotype and K-convexity, This Handbook.
D. Preiss, Geometric measure theory in Banach spaces, This Handbook.
H.E Rosenthal, The Banach spaces C (K), This Handbook.
L. Tzafriri, Uniqueness of structure in Banach spaces, This Handbook.
P. Wojtaszczyk, Spaces of analytic functions with integral norm, This Handbook.
M. Zippin, Extension of bounded linear operators, This Handbook.
V. Zizler, Nonseparable Banach spaces, This Handbook.

CHAPTER 2
Positive Operators
Y.A. Abramovich
Department of Mathematical Sciences, Indiana University-Purdue University, Indianapolis, IN, USA
E-mail: yabramovich @ math. iupui, edu
C.D. Aliprantis
Department of Economics and Department of Mathematics, Purdue University, West Lafayette, IN, USA
E-mail: aliprantis @ mgmt.purdue, edu
Contents
1. Ordered vector spaces and Banach lattices .................................. 87
2. Operators between Banach lattices ...................................... 90
3. When is every continuous operator regular? ................................. 91
4. Domination, compactness, and factorization ................................. 93
5. Invariant subspaces of positive operators ................................... 98
6. Compact-friendly operators and invariant subspaces ............................. 102
7. Integral operators and invariant subspaces .................................. 105
8. Applications ................................................... 111
Acknowledgment .................................................. 117
Added in Proof ................................................... 117
References ..................................................... 117
HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 1
Edited by William B. Johnson and Joram Lindenstrauss
9 2001 Elsevier Science B.V. All fights reserved
85

Positive operators 87
The theory of positive operators is a distinguished and significant part of the field of
general operator theory. The extra feature of this part of operator theory is the existence
of an order on the spaces involved. This ingredient is of tremendous importance and lies
in the core of many specific results valid for positive operators. The two main objectives
of the present work are to discuss the relationships between general operators and positive
operators and to demonstrate the effects the order structure has on general operators acting
between Banach lattices.
We list here several books devoted primarily to positive operators between ordered
spaces: [15,22,29,41,75,78-80,86,87,96,100,123,126,135,144,145]. The size constraints
of this article as well as the areas of interest of the authors have inevitably influenced
the selection of the material for this survey and have precluded us from mentioning many
existing directions within the theory of positive operators. We especially regret having to
omit the ergodic and interpolation properties of positive operators. For these topics the
reader is referred to [83] and [84] respectively. Since our survey is devoted to operators,
we have reduced to a bare minimum the general theory of Banach lattices presented here.
Most of the books cited above, the series of notes by Luxemburg and Zaanen [93,92] and
the two surveys [39,40] are excellent sources for such a theory as well. Throughout this
work the word "operator" is synonymous with "linear operator".
1. Ordered vector spaces and Banach lattices
Though we will be dealing mostly with operators on Banach lattices, it is fruitful to start
with the general framework of (partially) ordered vector spaces. We will introduce some
necessary terminology and prove a few basic results; for a systematic presentation we refer
the reader to [70,94,101,107,134,143].
Recall that a real vector space X equipped with a partial order ~> is said to be a (partially)
ordered vector space whenever x ~ y imply otx ~> oty for all ol ~> 0 and x + z ~> y + z for
all z E X. The set X+ = {x E X: x ~> 0} is called the positive cone of X and its elements
are referred to as positive vectors. The cone X+ is said to be generating whenever X --
X+ - X+, i.e., whenever every vector can be written as a difference of two positive vectors.
An ordered vector space X is said to be Archimedean whenever nx 0 or x ~< 0. E]
THEOREM 2 (M. Krein-Smulian). Let X be a Banach space ordered by a closed generating
cone. Then there is a constant M > 0 such that for each x ~ X there are x l, x2 E X+
satisfying x -- xl - x2 and Ilxi II ~< Mllxll for each i.

88 Y.A. Abramovich and C.D. Aliprantis
PROOF. We present a sketch of the proof. For each n define the set
En = {x ~ X" =tXl, X2 G_ X- t- with x - Xl - x2 and [Ixi II ~ n (i = 1, 2) }.
Clearly, each En is convex, symmetric, and 0 6 En. In addition, note that En cc_ Em whenever
n ~< m.
Since X+ is generating, we see that X - Un~=l En. So, by the Baire Category Theorem,
some Ek contains a closed ball B(xo, r) = {x 6 X: Ilx0-xll ~< r}. The properties ofthe sets
En imply that B(0, r) c_ E~. Now by imitating the proof of the open mapping theorem, we
can show that B(O, r) c_ Ezk holds and the proof is done. The details can be found in [135];
see also [4].
D
COROLLARY 3. Let X be an ordered Banach space whose positive cone is closed and
generating. If xn --+ x holds in X, then there exist Yn, Zn, Y, z ~ X+ such that Xn -- Yn -- Zn,
x = y -- Z, Yn --+ Y and Zn --+ z.
COROLLARY 4. Let X be a Banach space partially ordered by a closed generating cone,
and let Y be a topological vector space. Then an operator T : X --+ Y is continuous if and
only if the restriction of T to X+ is continuous.
And now we introduce the central concept of this work.
DEFINITION 5. An operator T:X ~ Y between two ordered vector spaces is said to be
positive if T (X+) _ Y+, i.e., if x ~> 0 implies Tx >~ O.
It is a remarkable fact that quite often positive operators are automatically continuous.
This was first proven by M. Krein for positive linear functionals [81 ] and later was generalized
in several contexts by various authors; see, for instance, [32,99,101,121]. The next
result, due to Lozanovsky, is the strongest in this direction and appeared in [136].
COROLLARY 6 (Lozanovsky). Let X and Y be two ordered Banach spaces whose cones
are closed. If additionally, the cone of X is generating, then every positive operator
T : X --+ Y is continuous.
PROOF. It suffices to show that a positive operator T has a closed graph. So, assume
Xn --+ 0 in X and Txn --+ y in Y. By passing to a subsequence, we can also assume
oo
Y~n=l n ]lXn I] < cx~. By Theorem 2 there exist some M > 0 and two sequences {Yn } and
{Zn} of X+ satisfying Xn = Yn - Zn, Ilynl] ~< m[Ixnll, and I[Zn [] ~< mllxnl[ for each n. Since
X+ is closed, the vector z = ~-~n~=l n(yn + Zn) of X belongs to X+ and -z

Positive operators 89
A norm I1" II on a vector lattice is said to be a lattice norm if Ixl ~ lyl implies Ilxll ~ Ilyll.
A normed vector lattice is a vector lattice equipped with a lattice norm. A norm complete
normed vector lattice is called a Banach lattice. Here is a list of the most important proper-
ties between the topological and order structures of a Banach lattice that will be employed
in this work. The first one, the order continuity, is discussed at length in [72].
DEFINITION 7. A lattice norm on a vector lattice X is said to be:
(1) order continuous, if xc~ $ 0 implies ]]x~ 11 $ 0;
(2) a Levi norm, if 0 ~< x~ 1" and ]lx~ 11 ~< 1 imply that sup~ x~ exists in X;
(3) a Fatou norm, if 0 ~< xc~ 1" x implies Ilx~ II 1" Ilxll;
(4) a weak Fatou norm, if there exists some constant K ~> 1 such that
0 ~ xo~ 1" x ------->, IIx II ~ K lim Ilxo~ II.
Of
A Banach lattice with an order continuous Levi norm is usually referred to as a
Kantorovich-Banach space, or as a KB-space in short. The meanings of the expressions
"X is a Levi Banach lattice", or "X has the Levi property", or "X has the weak Fatou
property", etc. should be clear and they will be used throughout this work.
If X has a Levi norm, then obviously X is Dedekind complete. Note that two equivalent
(lattice) norms are either both Levi or both fail to be Levi. The same is true for the weak
Fatou property. It is well known [30] that if a norm is Levi, then necessarily it is also weak
Fatou. Property (3) is clearly an isometric property.
If, in the definition of an order continuous, or a Levi, or a (weak) Fatou norm one replaces
arbitrary nets by sequences, then one arrives at the definition of a sequentially order
continuous norm, or a sequentially Levi norm, or respectively, of a sequentially (weak)
Fatou norm. 1 Thus, for example, a norm on a Banach lattice X is sequentially Fatou (or
simply ~-Fatou) if for each sequence 0 ~< xn t x we have ]Ix ]1 = lim ]]xn 11.
There is a very rich structural theory of Banach lattices (reflecting partly the correspond-
ing theory of Banach spaces) in which various properties of Banach lattices are character-
ized by the absence or presence of subspaces or vector sublattices which are isomorphic to
some classical spaces ~p (1 ~< p ~< ec) or co. We refer to [29,39,40,96] for these results.
As an illustration, we mention only the famous theorem stating that a Banach lattice X is
a KB-space if and only if X does not contain a vector sublattice order isomorphic to co.
1 These properties appear in the literature under many different names. Fremlin [58] was the first to associate the
Levi and Fatou names with these properties. Nakano [100, pp. 129-130] used the term monotone complete norm
for a sequentially Levi norm and universally monotone complete norm for a Levi norm. Zaanen [144] considers
sequentially Levi norms in two places under different names. First on p. 305 he refers to them, like Nakano, as
monotone complete norms, and then on p. 421 as norms with the weak Fatou property for monotone sequences.
The term weak Fatou property for directed sets (p. 390) is used by Zaanen for what we refer to as a Levi norm.
Meyer-Nieberg [96, p. 96], uses the term "monotonically complete norm" for a Levi norm. The Soviet school on
Banach lattices used the symbols (A) and (A f) to denote the sequential order continuous and order continuous
norms, respectively; they also used the symbols (B) and (B r) to denote the sequential Levi and the Levi properties
respectively, and finally they used the symbols (C) and (C f) to denote the sequential Fatou and the Fatou properties
respectively.

90 YA. Abramovich and C.D. Aliprantis
A vector subspace J of a vector lattice X is said to be an (order) ideal if Ixl ~ lyl and
y 6 J imply that x 6 J. For each 0 0 such that Ix[ ~< )~u}.
A positive element u 6 X is called a (strong) unit if Xu = X. Each unit generates the so
called unit-norm II" Ilu via IIxllu = inf{)~ >~ 0: Ix[ ~< ~.u}. Clearly this norm is always Fatou;
if X is a-Dedekind complete, then I]" Ilu is complete.
2. Operators between Banach lattices
If X and Y are Banach spaces, then the symbol s Y) denotes the Banach space of all
continuous operators from X to Y; we let s = E(X, X). 2 If X and Y are in addition
Banach lattices, then there are several very important classes of operators associated with
the order structure on these spaces. We denote by 12+ (X, Y) the collection of all positive
operators. This collection is a cone and it induces a natural (partial) order on the Banach
space C(X, Y).
The subspace s Y) - E+(X, Y) of/2(X, Y) generated by the cone/2+(X, Y) is
denoted by/2 r (X, Y), or simply/2 r, and is referred to as the space of regular operators.
In other words, an operator is said to be regular if it can be written as a difference of two
positive operators.
An operator T" X --+ Y is order bounded if T maps order bounded sets in X to order
bounded sets in Y. The symbol/2 b (X, Y) denotes the space of all order bounded operators.
Clearly each regular operator is order bounded, that is, s c_ E b. The converse is not true in
general (see [29] for several counterexamples, the first of which was found by S. Kaplan).
Here we can ask the question: when does the equality s = F b hold? The next famous
theorem of E Riesz and L. Kantorovich describes a rather general situation when this
happens.
THEOREM 8 (Riesz-Kantorovich). Let X, Y be vector lattices with Y Dedekind complete.
Then E r (X, Y) = 12 b (X, Y), i.e., each order bounded operator T : X --+ Y is regular. Moreover,
in this case, Er (X, Y) is itself a Dedekind complete vector lattice whose lattice operations
satisfy the Riesz-Kantorovich formulas, according to which for each T ~ E r (X, Y)
and each x ~ X+ we have
T+(x) -- sup{Tu" 0

Positive operators 91
(2) Is there a non-Dedekind complete vector lattice Y such that s (X, Y) is a vector
lattice for each (some) vector lattice X? The answer is yes, and we refer to [18] for
such examples and related discussions.
(3) Assume that E r (X, Y) is a vector lattice, and so for each T E E r (X, Y) its modulus
I TI exists. Is it true that this operator is necessarily given by the Riesz-Kantorovich
formula? This is a long standing open problem.
(4) Assume that for a fixed vector lattice Y the vector space s (X, Y) is a vector lattice
for all vector lattices X in some special class of vector lattices. What impact does
this have on the order structure of Y?
In connection with (3) note that even when s (X, Y) is not assumed to be a vector
lattice, there is no single example in the literature of a regular operator T:X --+ Y for
which its modulus I TI exists and cannot be obtained by the Riesz-Kantorovich formula.
Abramovich and Wickstead have studied this question rather systematically and described
many cases covered by the Riesz-Kantorovich formula [ 18,19,140]. For the special domain
of C(K)-vector lattices, this problem was also studied by van Rooij [118,119].
Now let X and Y be Banach lattices. On the space E(X, Y) we will consider the usual
operator norm. Under this norm the subspace s (X, Y) is rarely closed. However, there
is a natural norm on s (X, Y), the so-called regular norm, which is closely related to the
structure of the regular operators. The regular norm on s (X, Y) is defined by
IlTllr - inf{ IISII" s E s
Y) and S ~> iT}.
Clearly II T II ~ IIT IIr and for positive operators both norms coincide. It is well known that
(Z; r (X, Y), II" IIr) is a Banach lattice. If Y is Dedekind complete, then the modulus Irl
exists by Theorem 8, and in this case II r lit = IlITI II.
3. When is every continuous operator regular?
As the title of this section indicates, we shall discuss here the pairs of Banach lattices X
and Y for which the space of all continuous operators E(X, Y) coincides with the space of
all regular operators s (X, Y). We should distinguish here between two closely related but
different questions. The first one requires simply the set-theoretic equality
IZ(X, Y) = ~,r (x, Y), (*)
while the second requires (,) and, additionally, the norm equality
IIT II = II T Ilr for each T E L;(X, Y). (**)
We will refer to these problems as the isomorphic and isometric problems, respectively.
The following two theorems, which are essentially due to Kantorovich and Vulikh [76],
describe the two most important classes of Banach lattices for which each continuous operator
is regular. These are the familiar AM- and AL-spaces, also known under the names
abstract M spaces and abstract L l spaces, respectively, and defined in [72, Section 5].

92 Y.A. Abramovich and C.D. Aliprantis
THEOREM 9. Let Y be a Dedekind complete Banach lattice with a strong unit. Then for
each Banach lattice X equality (,) holds, that is, s Y) = s (X, Y). If the norm on Y
is Fatou (in particular, if Y is equipped with a unit norm and, thus, is an AM-space), then
(**) also holds.
THEOREM 10. If X is an AL-space and Y is an arbitrary Banach lattice with a Levi norm,
then equality (,) holds. If the norm on Y is Fatou, then we also have (**).
To be precise, it was assumed in [76] that Y in Theorem 10 was a KB-space, and it was
noticed in [ 129] that the original proof could be easily carried over to an arbitrary Banach
lattice with a Levi norm. For a KB-space Y the proofs can be found in [29, Theorem 15.3],
and [ 134, Theorem 8.7.2]. Under the assumption that Y is positively complemented in Y**
(which is somewhat stronger than the Levi property) a proof of Theorem 10 is presented
in [96, Theorem 1.5.11]. Two principal questions can be asked in connection with the
previous theorems.
(1) Do Theorems 9 and 10 characterize the AL- and AM-spaces?
(2) To what extend is the Levi condition essential for the validity of (,)?
The first question is quite old and has various interpretations depending on whether we
deal with the isometric or the isomorphic version. We refer to [3] for a brief survey of
major results in this direction. Here we mention a few of them, starting with the isometric
version of this question.
THEOREM 1 1. Suppose that the Banach lattices X and Y satisfy both (,) and (**). Then
either X is an AL-space or Y is an AM-space.
The isomorphic version is much deeper and has, in general, a negative solution [2].
THEOREM 12. There exist two Dedekind complete Banach lattices X and Y satisfying (,)
such that X is not order isomorphic to any AL-space and Y is not order isomorphic to any
AM-space. Moreover, for any given e > 0 the Dedekind complete Banach lattices X and Y
can be constructed in such a way that IIT IIr ~< (1 + e)II T ]l for each T ~ 12(X, Y).
The strongest affirmative isomorphic result is also due to Cartwright and Lotz [44] and is
stated next. A simple proof of this result is given in [ 17], where the technique introduced by
Tzafriri in [133] of abstract Rademacher functions is utilized. The works of Fremlin [59],
~rno [ 105] and Schaefer [ 122] contained some versions of the next result.
THEOREM 13. Let two Banach lattices X and Y satisfy (,), and assume that either X*
or Y contains vector sublattices uniformly in n isomorphic to g n p for some p E [1 cx~). In
the former case Y is order isomorphic to an AM-space and in the latter case X is order
isomorphic to an AL-space.
The case when X = Y is exceptionally nice and was studied in [2].
THEOREM 14. If a Banach lattice X satisfies s
either to an AL- or AM-space.
= s (X), then X is order isomorphic

Positive operators 93
PROOF. To sketch a proof, note that the following alternative is true: X either contains
uniformly in n the vector sublattices order isomorphic to g~ or it does not. In the former
case, X* contains uniformly in n the vector sublattices g~ and so, by Theorem 13, X is
isomorphic to an AM-space. In the latter case, X is isomorphic to an AL-space by Drno's
theorem. []
Questions (1) and (2) formulated above, though quite natural, have been addressed by
Abramovich and Wickstead only recently in [21 ].
THEOREM 15. A Dedekind complete Banach lattice Y has a Levi norm if and only if for
every AL-space X we have s Y) = s (X, Y).
This converse to Theorem 10 is somewhat partial since we assume Y to be Dedekind
complete. On the other hand, as the next two theorems in [21 ] show, it is the best one can
get. Note also that there are non-Dedekind complete Banach lattices (hence, without a Levi
norm) which, nevertheless, satisfy (,).
THEOREM 16. For a Banach lattice Y the following statements are equivalent:
(a) Y is Dedekind complete.
(b) For all Banach lattices X, the space f_r (X, Y) is a Dedekind complete vector lattice.
(c) For all AL-spaces X, the space s (X, Y) is a vector lattice.
THEOREM 17. For a Banach lattice Y the following statements are equivalent:
(1) Y is ~-Dedekind complete.
(2) s (X, Y) is ~-Dedekind complete for each separable Banach lattice X.
(3) s (c, Y) is a vector lattice, where c is the space of all convergent sequences.
(4) /:r (LI [0, 27r], Y) is a vector lattice.
We close this section by mentioning one more direction of research devoted to the connections
between the order and topological properties of compact operators. Namely, what
can be said about the modulus of a compact operator T :X --+ Y between Banach lattices?
Does this modulus exist? When is it compact or weakly compact? A well known example
by Krengel [29, Example 16.6] shows that even on ~2 the modulus of a compact operator
may fail to exist. We refer to [20] for some history, further references and concrete results
concerning the properties of the modulus of a compact operator.
4. Domination, compactness, and factorization
DEFINITION 18. Let T, B : X --+ Y be two operators between vector lattices with B positive.
We say that the operator T is dominated by the operator B (or that B dominates T)
provided ]T(x)l ~< B(Ixl) for each x 6 X.
It should be clear that every operator between Banach lattices dominated by a positive
operator is automatically continuous, and that a positive operator T is dominated by another
positive operator B if and only if 0 ~< T ~< B. When Y is Dedekind complete, an

94 Y.A. Abramovich and C.D. Aliprantis
operator T is dominated by a positive operator B if and only if T is regular and I TI ~< B
holds (use Theorem 8 to check this).
We are now ready to discuss a new and interesting direction dealing with dominated
operators on Banach lattices. This direction can be described collectively as the domination
problem.
THE DOMINATION PROBLEM. Assume that B : X --+ Y is a positive operator between Banach
lattices and assume that B satisfies some property ( P ). What effect does this property
have on an operator T dominated by B ?
Of course, the best one can expect is that T also satisfies (P). Here are a few examples
of the properties that have been studied in connection with the domination problem:
compactness, weak compactness, Dunford-Pettis, and Radon-Nikodym.
A first general domination result was established in 1972 by Abramovich [ 1 ] for weakly
compact operators. He proved that if a linear operator S : X --+ Y from a Banach lattice to
a KB-space is dominated by a weakly compact operator, then S is also weakly compact.
Several years later Pitt [ 109] proved a domination result for compact operators, namely he
showedthatif B isacompactpositive operatoron Lp(#) with 1 < p < oo andO

Positive operators 95
The exponents three and two in the previous two theorems are the best possible. Some
historical comments regarding this theorem and its predecessors can be found in [29,
p. 279]. Are there any other cases (not covered by the Dodds-Fremlin Theorem) where
one can deduce the compactness of the dominated operator itself? It turns out that there
are. This was shown by Wickstead in [142]. Before stating this result, let us remind the
reader that by a well known theorem of Walsh [ 137] the atomic Banach lattices with order
continuous norm are precisely those Banach lattices in which every order interval is norm
compact, and that in such spaces the solid hull of any norm compact set is again norm
compact [138, Theorem 5].
THEOREM 22. For a pair of Banach lattices X and Y the following two statements are
equivalent:
(1) Every operator from X to Y which is dominated by a compact operator is itself
compact.
(2) One of the following three (non-exclusive) alternatives holds:
(a) Both X* and Y have order continuous norm.
(b) Y is an atomic Banach lattice with order continuous norm.
(c) X* is an atomic Banach lattice with order continuous norm.
A complete solution of the domination problem for weakly compact operators preceded
the solution of the compact case and was found also by Wickstead [ 139].
THEOREM 23. For a pair of Banach lattices X and Y the following two statements are
equivalent:
(1) Each operator from X to Y which is dominated by a weakly compact operator is
weakly compact.
(2) Either X* or Y has order continuous norm.
The following analogue of Theorem 21 for weakly compact operators (that is, when no
conditions on Banach lattices is imposed) was established in [26].
THEOREM 24. Suppose that in the scheme X M~ y M2 Z of continuous operators between
(real or complex) Banach lattices each operator Mi is dominated by a weakly compact
positive operator, then M2 Ml is a weakly compact operator. In particular, if an operator
T : X --+ X on a Banach lattice is dominated by a weakly compact operator, then T 2
is weakly compact.
Recall that a continuous operator T:X --+ Y between Banach spaces is said to be a
Dunford-Pettis operator if T carries weakly precompact subsets of X onto norm precompact
subsets of Y. The domination problem for Dunford-Pettis operators was originally
considered by Aliprantis and Burkinshaw [27]. Improving the main result in [27], Kalton
and Saab [73] proved the following theorem.
THEOREM 25. Let B : X --+ Y be a Dunford-Pettis operator between Banach lattices and
assume that Y has order continuous norm. Then any operator T dominated by B is also
Dunford-Pettis.

96 Y.A. Abramovich and C.D. Aliprantis
Wickstead [ 142] showed that the only other case when the Dunford-Pettis property is
inherited by a dominated operator is when X has weakly sequentially continuous lattice
operations. The atomic Banach lattices with order continuous norm and the AM-spaces
have this property. For such X the Banach lattice Y may be arbitrary. Some interesting
results on the domination problem for nonlinear operators can be found in [38].
Now we turn our attention to the factorization problem. Recall that a continuous operator
T : X --+ Y between Banach spaces factors through a Banach space Z if there exist
continuous operators U:X --+ Z and V:Z --+ Y such that T -- V U. Depending on the
properties of the space Z and of the operators U and V, a lot can be learned about the
properties of the operator T. If the spaces under consideration are Banach lattices, then we
can immediately look at positive or regular operators between them and demand more of
the factorization space Z and of the factor operators U and V.
A classical example to illustrate this topic is the following. If a continuous operator
T:X --+ Y between Banach spaces factors through a reflexive Banach space, then T is
weakly compact. Remarkably, as shown by Davis, Figiel, Johnson, and Petczyfiski [50],
this is a characterization of weakly compact operators.
THEOREM 26. An operator between two Banach spaces is weakly compact if and only if
factors through a reflexive Banach space.
As a matter of fact, Theorem 26 can be considered as the starting point for the subject
of factorization. Without trying to be complete on this topic, we will present here only a
few results on factorization of operators between Banach lattices. Additional material on
factorization of positive operators may be found in [28] and [96].
Figiel [57] and Johnson [71 ] found which extra conditions on the factors and the transient
space can be deduced for compact operators.
THEOREM 27. Every compact operator between Banach spaces factors with compact factors
through a separable reflexive Banach space.
Inspired by these results, Aliprantis and Burkinshaw [28] studied the factorization problem
under the presence of order structures. The following question is central for this topic.
THE FACTORIZATION PROBLEM. Does every positive compact operator acting between
Banach lattices factor (with positive compact factors if possible) through a reflexive Banach
lattice?
This problem (reiterated in [68] as Problem 9) is still open. We will mention some recent
affirmative results below. But first, we should point out that a slightly weaker problem,
when the given operator is assumed to be only weakly compact, is very difficult, too. It has
been solved in the negative not a long time ago by Talagrand [130].
EXAMPLE 28. There exists a positive weakly compact operator T:s
cannot be factored through a reflexive Banach lattice.
--~ C[0, 1] which

Positive operators 97
The role of the space C[0, 1], or more generally, of the spaces C(K), for the factorization
problem was studied in great detail by Ghoussoub [61 ] and Ghoussoub and Johnson in [62,
63]. On a positive note regarding the factorization problem, we will mention the following
theorem obtained in [28].
THEOREM 29. The square of a positive compact (respectively weakly compact) operator
on a Banach lattice factors through a reflexive Banach lattice with positive compact
(respectively weakly compact) factors.
From this theorem and Example 28 it follows that not each weakly compact operator
is the square of a weakly compact operator. This, of course, follows from other known
results as well. (For instance, there are cases when the square of a weakly compact operator
is necessarily compact.) The next result by Aliprantis and Burkinshaw is a special
case of Theorem 17.13 in [29]; it demonstrates a nice relation between factorization and
domination.
THEOREM 30. Assume that T : X --+ X is a positive compact operator on a Banach lattice
whose dual X* has order continuous norm. Then there exist a reflexive Banach lattice Z
and positive operators U : X --+ Z, V : Z --+ X such that T = V U. Moreover, any operator
S : X --+ X which is dominated by the operator T also factors through Z and S -- V1U for
some O

98 YA. Abramovich and C.D. Aliprantis
An example in [110] shows that one cannot improve (3) by getting, for a positive regularly
approximable T, both factors positive. A detailed study of when, in that example,
both factors can be chosen positive is presented in [ 111 ].
5. Invariant subspaces of positive operators
In this section we will discuss the Invariant Subspace Problem for operators that are either
positive or closely associated with positive operators. The general theory concerning the
invariant subspace problem will be presented in a separate article prepared for this volume
by Enflo and Lomonosov.
THE INVARIANT SUBSPACE PROBLEM. Does a continuous linear operator T : X --+ X on
a Banach space have a non-trivial closed invariant subspace?
A vector subspace is "non-trivial" if it is different from {0} and X. A subspace V of X is
T-invariant if T (V) _c V. If V is invariant under every continuous operator that commutes
with T, then V is called T-hyperinvariant.
If X is a finite dimensional complex Banach space of dimension greater than one,
then each non-zero operator T has a non-trivial closed invariant subspace. On the other
hand, if X is non-separable, then the closed vector subspace generated by the orbit
{x, Tx, TZx .... } of any non-zero vector x is a non-trivial closed T-invariant subspace.
Thus, the "invariant subspace problem" is of substance only when X is an infinite dimensional
separable Banach space. Accordingly, without any further mentioning, all Banach
spaces under consideration in this section will be assumed to be infinite dimensional separable
real or complex Banach spaces. The only exception will be made while discussing
the Perron-Frobenius Theorem.
In 1976, Enflo [56] was the first to construct an example of a continuous operator on
a separable Banach space without a non-trivial closed invariant subspace, and thus he
demonstrated that in this general form the invariant subspace problem has a negative answer.
Subsequently, Read [ 115-117] has constructed a class of bounded operators on el
without invariant subspaces. For operators on a Hilbert space, the existence of an invariant
subspace is still unknown and is one of the famous unsolved problems of modern mathematics.
Due to the above counterexamples, the present study of the invariant subspace
problem for operators on Banach spaces has been focused on various classes of operators
for which one can expect the existence of an invariant subspace.
We start our invariant subspace results with a version of the classical Perron-Frobenius
theorem for positive matrices. As usual, we denote by r(T) the spectral radius of an operator
T.
THEOREM 3 2 (Perron-Frobenius). If A is a non-negative n x n matrix such that for some
k >~ 1 the matrix A k has strictly positive entries, then the spectral radius of A is a strictly
positive eigenvalue of multiplicity one having a strictly positive eigenvector.
The proof of this theorem, discovered by Frobenius [60] and Perron [108], is available
in practically every book treating non-negative matrices, for instance in [33,35,98]. One

Positive operators 99
more proof of the Perron-Frobenius theorem as well as many interesting generalizations
1
can be found in [ 112]. If all entries of A are strictly positive, then the sequence { [r(a)] k Aku}
converges to a unique strictly positive eigenvector corresponding to the eigenvalue r (A), no
matter which initial vector u > 0 is chosen. This fact has numerous applications. A major
step in extending the Perron-Frobenius Theorem to infinite dimensional settings was done
by Krein and Rutman [82] who proved the following theorem.
THEOREM 3 3 (Krein-Rutman). For any positive operator T" X --~ X on a Banach lattice
r(T) E or(T), i.e., the spectral radius of T belongs to the spectrum of T. Furthermore, if
T is also compact and r(T) > O, then there exists some x > 0 such that Tx -- r(T)x.
PROOF. We will sketch a proof. The inclusion r(T) E cr(T) is caused merely by the positivity
of T. Indeed, if we denote by R()~) the resolvent operator of T, then clearly R0~) > 0
for each )~ > r(T), Also for each )~ with 1)~1 > r(T) the inequality ]]R(])~[)]] ~> ]]R()~)]]
1
holds. Therefore, for )~, -- r(T) + -d we have []R()~n)l[ ~ oc, whence r(T) ~ or(T).
Assume further that T is compact and r(T) > 0. There exist unit vectors Yn E X+
such that ]lRO~,)yn I[--~ oc. Using the vectors y,, we introduce the unit vectors xn =
R()~n)yn/llR()~n)Yn][ ~ X+. Since T is compact we can assume that Txn --+ x ~ X+. Finally,
using the identities
r(T)xn - Txn -
[r(T)- Xn]xn + ()~n - T)xn
= [r(T) - ~,]x, + y~/IIRO~)y~II
and that r(T) > 0 we infer that x 5~ 0 and that Tx -- r(T)x.
The conclusion of the previous theorem remains valid if we replace the compactness
of T by the compactness of some power of T. Indeed, assume that T k is compact for
some k. Since r(T k) = [r(T)] k > 0 the previous theorem implies that there is a vector
x > 0 such that Tkx = [r(T)]kx. It remains to verify that the non-zero positive vector
Y -- Y~i=0 k-I r iTk-l-i x is an eigenvector of T corresponding to the eigenvalue r(T).
The reader is referred to [121,123,144] for complete proofs and many pertinent results
concerning the Krein-Rutman theorem. Some relevant results can be found in [4]. Note
that the Krein-Rutman theorem holds not only for Banach lattices but for ordered Banach
spaces as well. There is an interesting approach allowing to relax the compactness assumption.
Namely, as shown by Zabrel3cO and Smickih [146] and independently by Nussbaum
[ 102], instead of the compactness of T it is enough to assume only that the essential
spectral radius re(T) is strictly less than the spectral radius r(T). A different type of relaxation
is considered in [121], where the restriction of T to X+ is assumed to be compact,
that is, T maps the positive part of the unit ball into a precompact set. A version of this
result, given in terms of re(T), can be found in [ 102].
Another classical result by M. Krein [82, Theorem 6.3] is the following.
THEOREM 34 (Krein). Let T" C(I-2) --+ C(I-2) be a positive operator, where S72 is a compact
Hausdorff space. Then T*, the adjoint of T, has a positive eigenvector corresponding
to a non-negative eigenvalue.

100 Y.A. Abramovich and C.D. Aliprantis
PROOF. Consider the set G -- {f E C(t'2)+ f(1) = 1}, where 1 denotes the constant
function one on t'2. Clearly, G is a nonempty, convex, and w*-compact subset of C(~)*.
Next, define the mapping F'G --+ G by
f+T*f f+T*f
F(f) -- =
[f + T'f](1) 1 + f(rl)
A straightforward verification shows that F indeed maps the set G into itself and that
F: (G, w*) --+ (G, w*) is a continuous function. So, by Tychonoff's fixed point theorem
(see, for instance [22, Corollary 16.52]) there exists some q~ E G such that F(4~) = 4~. That
is, q~ + T*4~ -- [1 + 4~(T1)]4), or T*4) -- 4)(T1)4~, establishing that 0 < 4~ E C(12)~_ is an
eigenvector for T* having the non-negative eigenvalue 4~(T1).
M
A proof of Theorem 34 that does not use fixed point theorems can be found in [77].
COROLLARY 3 5. Every positive operator on a C (Y2)-space (where Y2 is Hausdorff, compact
and not a singleton) which is not a multiple of the identity has a non-trivial hyperinvariant
closed subspace.
PROOF. Let T'C(S2) --+ C(Y2) be a positive operator which is not a multiple of the identity.
By Theorem 34 the adjoint operator T* has a positive eigenvector. If )~ denotes the
corresponding eigenvalue, then the subspace (T - )~I)(X) has the desired properties. V]
Recall that a continuous operator T : X ~ X on a Banach space is said to be quasinilpotent
if its spectral radius is zero. It is well known that T is quasinilpotent if and only if
limn~ ]lTnxl] 1/n = 0 for eachx c X. It can happen thata continuous operator T: X ~ X
is not quasinilpotent but, nevertheless, limn~ IlTnx ]ll/n = 0 for some x ~ 0. In this case
we say that T is locally quasinilpotent at x. This property was introduced in [6], where it
was found to be useful in the study of the invariant subspace problem. The set of points at
which T is quasinilpotent is denoted by QT, i.e., QT = {x 6 X: limn~ IlTnxll 1/n =0}.
It is easy to prove that the set QT is a T-hyperinvariant vector subspace. We formulate
below a few simple properties of the vector space QT.
9 The operator T is quasinilpotent if and only if QT = X.
9 QT = {0} is possible every isometry T satisfies QT = {0}. Notice also that even
a compact positive operator can fail to be locally quasinilpotent at every non-zero
vector9 For instance, consider the compact positive operator T:g2 -+ ~2 defined by
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
and note that IlTnxl] 1/n ~ ~]xk[ 1/n for each n, from which it follows that T is not
quasinilpotent at x.
9 QT can be dense without being equal to X. For instance, the left shift S: g2 -+ g2, defined
by S(xl, x2, x3 .... ) = (x2, x3 .... ), has this property.
9 If QT ~: {0} and QT ~ X, then QT is a non-trivial closed T-hyperinvariant subspace
of X.
The above properties show that as far as the invariant subspace problem is concerned,
we need only consider the two extreme cases: QT = {0} and QT = X.

Positive operators 101
We are now ready to state the main result about the existence of invariant subspaces
of positive operators on gp-spaces. It implies, in particular, that if a positive operator is
quasinilpotent at a non-zero positive vector, then the operator has an invariant subspace.
This is an improvement of the main result in [6].
THEOREM 36. Let T : g~p --+ g~p (1

102 YA. Abramovich and C.D. Aliprantis
A very interesting open problem related to Corollary 39 is whether or not each positive
operator on el has a nontrivial closed invariant subspace. Since each continuous operator
on el has a modulus (see Theorem 10), a natural candidate to test this problem is the
modulus of any operator on g l without a nontrivial closed invariant subspace. In particular,
each operator on g 1 without invariant subspace constructed by Read [ 115,117] is such a
candidate. Troitsky [ 131 ] has recently managed to handle the case of the quasinilpotent operator
T constructed by Read in [ 117]. Not only does I TI have a nontrivial closed invariant
subspace, but I TI also has a positive eigenvector.
In our previous discussion, we were considering only operators on gp-spaces. However,
we only used the discreteness of g p-spaces, the above results remain true for operators
on arbitrary discrete Banach lattices, 3 in particular, for operators on Lorentz and Orlicz
sequence spaces. For instance, the following analogue of Theorem 36 is true.
THEOREM 40. Let T : X --+ X be a continuous operator with modulus, where X is a discrete
Banach lattice. If there exists a non-zero positive operator S : X -+ X which is locally
quasinilpotent at a non-zero positive vector and SITI

Positive operators 103
the next two results. The proof of the first one was inspired by Hilden's proof (included
in [97]) of a simplified version of Lomonosov's invariant subspace theorem mentioned
above.
THEOREM 42. Let B : X --+ X be a positive operator on a Banach lattice. Assume that
there exists a positive operator S : X --+ X such that S B

104 Y.A. Abramovich and C.D. Aliprantis
where K0 is compact. Then the required "triplet" (R, C, K) is the following: R = B, C =
B and K = K0.
The proof that each positive integral operator is compact-friendly will be given in Theorem
52, and the proof for composition operators can be found in [ 11 ].
We will also need the following refinement of the concept of the strong unit introduced
at the end of Section 1. A positive element u in a Banach lattice X is called a quasi-interior
point whenever Xu is norm dense in X, i.e., Xu -- X.
LEMMA 45. If u is quasi-interior point in a Banach lattice X, then
(1) For every non-zero element y ~ Xu there exists an operator V : X ~ X dominated
by the identity operator and such that Vy > O.
(2) For every element v satisfying 0 0. Put
M1 -- VI C, and note that M1 is dominated both by the compact positive operator K and
by the operator R.
From Jx2 = X and C ~-0, we see that there exists some 0 < y

Positive operators 105
ICyl 0 and which is dominated by both
the positive compact operator K A and by RA.
From M3 M2 M1 x 1 : M3x3 > 0, we see that M3 M2 M1 is a non-zero operator which (by
Theorem 20) is also compact. Moreover, an easy argument shows that for each x 6 X
IM3M2M1 (x)[ ~ RARAR(Ixl) ~ [RARAR + T](Ixl).
Now consider the non-zero positive operator S -- RARAR + T. Then B and S commute,
S dominates the non-zero compact operator M3MzM1, and B is quasinilpotent at
x0. By Theorem 43, S and B have a common non-trivial closed invariant ideal. This ideal
is invariant under both B and T. [2
We will illustrate this theorem while studying integral operators in the next section. Theorem
46 may suggest that the compact-friendly operators characterize the positive operators
with an invariant subspace. However, this is not true. An example of a positive operator
that is not compact-friendly but has a non-trivial invariant subspace is given in [ 11 ]. The
operator itself is very simple, just a multiplication operator M~; but the fact that for some
multipliers the operator is not compact-friendly is far from being trivial. As shown in [ 13],
a multiplication operator M~o on an Lp(#) is compact-friendly if and only if the multiplier
q) is constant on a set of positive measure. Nevertheless, in the last section of [ 11],
a program is outlined for how the compact-friendly operators may be used to include the
exceptional cases as well.
We conclude this section by mentioning the following interesting conjecture of Lomonosov
[89]: The adjoint of an arbitrary operator on a Banach space has a non-trivial
closed invariant subspace. For operators on Banach lattices this conjecture is also open.
We refer to the papers by de Branges [36], Abramovich, Aliprantis and Burkinshaw [10]
and Simoni6 [127,128] that deal with this and related problems.
7. Integral operators and invariant subspaces
After a brief introduction to the theory of integral operators, we will prove in Theorem 52
that each positive integral operator is compact-friendly. After that, applying the theory
of compact-friendly operators, we will establish in Theorem 54 the existence of a nontrivial
invariant subspace for each locally quasinilpotent positive integral operator. The
rest of the section will be devoted to a discussion of the And6-Krieger theorem and its
generalizations.
We begin by defining the classes of integral (= kernel) and almost integral operators. 5
Throughout this section X and Y will denote two Dedekind complete Banach lattices which
5 To avoid a possible ambiguity let us point out that the integral operators considered in this section should not
be confused with the integral operators in the sense of Grothendieck discussed in [72].

106 Y.A. Abramovich and C.D. Aliprantis
are order dense ideals in the spaces (of equivalence classes) of all measurable functions
L0(I2j, El, #1) and L0(f22,272, #2) respectively, where (I21,271, #1) and (Y22, I72, #2)
are two arbitrary "non-pathological" measure spaces, for instance, a-finite measure spaces.
DEFINITION 47. An operator T" X --+ Y is said to be an integral operator if there exists
a #1 x #z-measurable function T (., .) on s x ~(22 such that for each x e X we have
Tx(t) -- f~ T(s,t)x(s)dlzl(s)
for #2-almost all t e S22.
1
The function T(., .) is usually referred to as the kernel of the operator T. It is easy
to prove that an integral operator is positive if and only if its kernel is nonnegative almost
everywhere on S21 x I22. Furthermore, it is well known that an integral operator T
with kernel T (., .) is a regular operator from X to Y if and only if for each x e X the
function f IT(s, .)llx(s)l d#l (s) belongs to Y. Moreover, under this condition the modulus
[TI:X --+ Y is also an integral operator, and its kernel is given by the absolute value
IT (s, t)[ of the initial kernel. It should be pointed out that many authors include the last
condition in the definition of the integral operator, thus guaranteeing that the modulus I TI
of the operator T also acts from X to Y. To stress this property, such operators are sometimes
referred to as absolute integral operators.
We refer to [41], [74, Chapter 11] and [144, Chapter 13] for a systematic study of integral
operators with emphasis on the order structure of the spaces involved. Apart from these
three books there are, of course, many other books devoted to integral operators; see for
instance [51,54,67,75,78]. Whenever convenient we will identify an integral operator with
its kernel.
The integral operators are closely related to finite-rank operators. A rank-one operator
from a Banach lattice E to a Banach lattice F is any operator of the form 4~ @ Y, where 4~ e
E*, y e F and ~b | y (x) -- 4~ (x)y for each x e E. Any operator of the form Y~i=ln ~bi | Yi
is called a finite-rank operator. We denote by E* | F the vector space of all finite rank
operators from E to F. The finite-rank operators are precisely the continuous operators
whose ranges are finite dimensional.
Recall that the subspace of all order continuous linear functionals on E is denoted by E*.
DEFINITION 48. A bounded operator T'E ~ F between Banach lattices, with F
Dedekind complete, is said to be an almost integral operator if T belongs to the band
generated in Z2 r (E, F) by the collection E* | F of all finite-rank operators.
The term "almost integral operator" has its origin in Theorem 50 below that is due to
Lozanovsky [90], who introduced both this class of operators and the term itself. According
to this theorem the classes of almost integral and integral operators are essentially the same.
Let us mention also the paper by Synnatzschke [129], where a refined version of almost
integral operators is studied. Namely, one can fix arbitrary subsets N _ E* and M c_ F,
and then define the band of almost integral operators (which depends on N and M) as the

Positive operators 107
band generated in/~r (E, F) by {q) | y" q9 E N, y E M}. The case of N = E* and M -- F
considered in Definition 48 is the most important.
A key element to Lozanovsky's characterization of almost integral operators is the famous
Dunford theorem (see Theorem 49) regarding the integral representation of operators
from L1 into Lp [55]. There are many proofs of this important theorem. A simple proof,
utilizing the work of Benedek and Panzone [34] on spaces with mixed norm, was also
found by Lozanovsky [91 ]. To outline the proof we need some terminology and notation.
For p E (1, ec), let q denote its conjugate exponent, that is, p-1 + q-1 _ 1. We denote
by L(p, oc) and L(q, 1) the usual Banach spaces with mixed norm [34], which consist of
all real-valued #l x #2-measurable functions f(s, t) on I21 x I22 satisfying respectively
the following conditions:
and
Ilfllp,oc "-Illlfllpll~- esssup
s ES22 1
If(s,t)lPd#2(t)
]'J'

108 Y.A. Abramovich and C.D. Aliprantis
THEOREM 50 (Lozanovsky). An operator T : X--+ Y between two Banach function
spaces is an almost integral operator if and only if T is a regular integral operator Furthermore,
for this operator there are measurable functions wi ~ Lo(#i) (i = 1, 2) and a
unique (up to an equivalence) integrable function T(s, t) on 121 x ~C22 such that for each
x ~ X and for lzz-almost all t ~ ~-22 the following representation holds:
Tx(t) = w2(t) fs2 T(s, t)x(s)wl (s) d/zl (s).
1
The idea of the proof lies in consecutive approximations obtained by "squeezing" X and
Y between weighted L~ and L1 spaces (this procedure is described in [72, Section 5]) and
then applying the Dunford theorem.
The previous theorem is a global order-theoretic characterization of the space of integral
operators. An important internal characterization of an individual integral operator is due
to A. Bukhvalov [37,41], see also [125], and it is given next. Many applications of this
criterion can be found in [41 ].
THEOREM 51. An operator T : X --+ Y between two Banach function spaces is an integral
operator if and only if for every order bounded sequence {Xn } C X which converges to zero
in measure, Txn ~ 0 holds #2-almost everywhere.
THEOREM 5 2. Every positive integral operator T is compact-friendly.
PROOF. As we already know T is an almost integral operator. Now note that for each
almost integral operator there exists a net { T~ } of positive operators such that 0 ~< T~ x t Tx
holds for each x ) 0, and each Tc~ is dominated by a positive finite-rank operator. This
implies that T is compact-friendly.
D
LEMMA 5 3. If S : X --+ X is a strictly positive integral operator on an order complete
Banach lattice, then for each element xo > 0 there exists a compact positive operator
K: X --+ X satisfying 0 ~ K ~ S 3 and K xo > O.
PROOF. Let S:X ---> X be a strictly positive integral operator and fix x0 > 0. So, there
exists a net { S~ } of positive operators such that 0 0, we see that there exists some index ~l such that S~x0 > 0
for each c~/> ~l. Similarly, from S~(S~,xo) ~ S(S~,xo) and S(S~,xo) > 0, we see that
S~(S~xo) > 0 for all ~ t> Ot2 ) Otl. Finally, from the facts that S(S~2S~lXO) > 0 and
S~(S~2S~lXO) t~ S(S~2S~,xo), it follows that S~(S~2S~,xo) > 0 for all possible indices
O~ /) O~ 3 /) Ot2 )/Otl. It remains to note that if we let K -- $33 , then K" X ~ X is a positive
compact operator satisfying 0 ~< K ~< S 3 and K xo > O.
And now we are ready to present one of the main invariant subspace theorems for integral
operators obtained in [7]. The proof illustrates the role of compact-friendly operators.
D

Positive operators 109
THEOREM 54. Let S : X --+ X be a non-zero positive integral operator on a Dedekind
complete Banach lattice and let B : X --+ X be another non-zero positive operator commuting
with S. If either S or B is locally quasinilpotent at a non-zero positive vector, then
the operators S and B have a common non-trivial closed invariant ideal.
PROOF. If the operator S is itself locally quasinilpotent at a non-zero positive vector, then
S is a compact-friendly operator that is locally quasinilpotent. So Theorem 46 is applicable
and guarantees the existence of the desired invariant subspace.
Assume now that B is locally quasinilpotent at a non-zero positive vector. Without loss
of generality we can assume that S is strictly positive (since otherwise the null ideal Ns
provides at once a common invariant subspace). Therefore, by Lemma 53, the operator S 3
dominates a non-zero compact positive operator K. Consider now the operator B 4- S 4- S 3 .
Clearly, it dominates K and commutes with both S and B. It follows that B is compactfriendly
and by hypothesis B is locally quasinilpotent at a non-zero positive vector. Hence,
Theorem 46, applied to B and B 4- S 4- S 3, guarantees the existence of a common nontrivial
closed invariant ideal for these two operators. Since both S and B are dominated by
B 4- S + S 3, it follows that this ideal remains invariant under both S and B.
D
COROLLARY 55. Every positive integral operator that is locally quasinilpotent at a nonzero
positive vector has a non-trivial closed invariant ideal.
For the classical Lp-spaces (and, as a matter of fact, for any Banach function space), the
preceding corollary yields the following.
COROLLARY 56. Let B : Lp(#) --+ Lp(~) (where # is a-finite and 1

110 Y.A. Abramovich and C.D. Aliprantis
We conclude this section by discussing the well known And6-Krieger theorem and its
generalizations.
DEFINITION 58. A positive operator T :X --+ X on a Banach lattice is ideal irreducible,
if T does not have a non-trivial invariant closed ideal, i.e., if T (J) ___ J and J is a closed
ideal, then either J = 0 or J = E. Similarly, T is band irreducible, if Tdoes not have a
non-trivial invariant band.
In 1957, And6 [31 ] proved that every band irreducible compact positive integral operator
has a strictly positive spectral radius. Twelve years later, in 1969, Krieger [85] removed
the compactness condition from And6's theorem, and the remarkable result asserting that
a band irreducible positive integral operator has a strictly positive spectral radius, is today
known as the And6-Krieger theorem. It is obvious that this result is an invariant subspace
result as soon as it is reformulated as follows: each quasinilpotent positive integral operator
has a non-trivial invariant band.
The proofs presented by And6 and Krieger were not at all transparent and several attempts
were made to clarify the intrinsic properties behind the structure of kernel operators.
In addition to these efforts, many investigators looked at the possibility of generalizing the
And6-Krieger theorem. In spite of all these research activities, it was not until somewhat
recently that the And6-Krieger theorem was understood relatively well and generalized in
various directions. Several people have contributed to the understanding of this subject;
among them Caselles [45,46], Grobler [65,66], de Pagter [106], Schaefer [124,120], and
Zaanen [ 144, Section 136].
The early attempts to generalize the And6-Krieger theorem to operators more general
than integral were fraught with many obstacles from the outset. Schaefer [ 120] constructed
an example of an ideal irreducible operator on Lp(#) (1 < p < e~) which was quasinilpotent.
This showed that even for the class of ideal irreducible operators (which is larger
than that of band irreducible operators) some extra condition is necessary. For a long time it
was not clear whether the compactness would be such a condition. This problem was solved
by de Pagter [ 106] who proved that every ideal irreducible compact positive operator on
a Banach lattice has a strictly positive spectral radius. This remarkable theorem was the
starting point in [5], where, instead of imposing all conditions on one operator, some of
the properties were shifted to the commutant of the operator. The next two results not only
demonstrate the difference between the ideal and band irreducible operators but also are
the strongest generalizations of the And6-Krieger theorem for ideal and band irreducible
operators obtained so far. We refer to [5] for these and many relevant results.
THEOREM 59. If an ideal irreducible positive operator U : X ~ X commutes with a compact
positive operator V : X --+ X, then r(UV) > O, and in particular r(U) > 0 and
r(V) > O.
PROOF. We will outline the main points of the proof. First of all, let us show that every
non-zero compact quasinilpotent positive operator T has a non-trivial closed positively
hyperinvariant ideal J, that is, J is invariant under each positive operator in the commutant
{T}' of T.

Positive operators 111
Let F be the closed ideal generated by the range of T. We claim that F is the closure
of a principal ideal. The compactness of T implies that the range of T is separable. So,
there exists a countable subset {yl, y2 .... } of T (X) consisting of non-zero vectors that is
norm dense in T(X). Consider u -- y~'n~__l 2,I~1'~I, ii which is in F. For the principal ideal
A, generated by u we clearly have A, _ F. Conversely, the inclusion {yl, y2 .... } _ A,
implies that T (X) ___ A,. Hence, F c_ Au and so A, = F.
Clearly, F 5~ {0} and it is easy to verify that F is indeed a positively hyperinvariant
ideal for T. We are done if F is proper. So assume that F -- A, = X, which implies in
particular that X has quasi-interior points. In this case, the existence proof of a non-trivial
closed positively hyperinvariant ideal is just the second part of the proof of de Pagter's
result mentioned above [106, Proposition 2]. The basic steps of this proof follow.
Let J+ = {0~< S E s 3 R E {T}'with0~< S ~< R}. Also, consider the vector
subspace ,7 = {S1 - $2: Sl, $2 E ,.7 + } of s In addition, for each x >~ 0, let
J[x] = {Sx: S E J}. It is shown in [106] that:
(1) For each x > 0 the closure J[x] is a non-zero T-invariant closed ideal; and
(2) For some x > 0 the closed ideal J[x] is non-trivial.
Next, let C be a positive operator that commutes with T. If S E J, then there exist
S1,82 E ,J+ and RI, R2 E {T}' satisfying S = Sl - 82 and 0 ~< Si ~ Ri (i -- 1,2). Then,
0

112 Y.A. Abramovich and C.D. Aliprantis
and details. Each of these applications requires the development of fairly sophisticated
machinery that cannot be presented here because of space limitations. More areas of applications
can be found in the interesting surveys by Dancer [49] (devoted to some recent
work on maps, both linear and nonlinear, between cones in Banach spaces) and by Nussbaum
[103,104] (devoted to the fixed point index and some of its applications).
Biology. We begin with some interesting and surprising applications of the abstract resuits
on domination described in Section 4. The first example of such an application was
discovered by Btirger [42] in biology. In [42] a very general discrete-time model that describes
the evolution of type densities under the action of mutation and selection in an
asexually reproducing population is developed and analyzed. As a result of this analysis,
the author proves the existence and uniqueness of a strictly positive equilibrium.
The analysis unfolds in some L1 (#)-space, that is, in a special Banach lattice with order
continuous norm and so the Aliprantis-Burkinshaw Theorem 21 is applicable. This
allows the author to conclude that for some positive compactly dominated integral operator
U its second power U 2 is compact and then to apply to U the improvement of
the Krein-Rutman theorem mentioned after Theorem 33, according to which r(U) > 0 is
an eigenvalue, and U has a unique strictly positive normalized eigenvector corresponding
to r(U). This model and its analysis have been recently generalized by BUrger and
Bomze [43].
A similar application, but for a completely different model, has been obtained by
Rhandi [ 113]. He has also applied Theorem 21 in his investigation of a population equation
to obtain compactness of the remainder of order 3 of the Dyson-Philips expansion. In his
recent work with Schnaubelt [114], they deal with a more complicated model describing
a non-autonomous population equation with diffusion in L1, and again apply Theorem 21
to deduce the compactness of some operators. This allows them to obtain existence and
uniqueness of positive solutions to this equation.
A beautiful application of the Perron-Frobenius Theorem 32 to simple growth processes
is given next. Suppose that there are N different types of biological objects. Assume also
that at the times t = 0, 1, 2 .... each individual object produces a certain number of offspring
among which every type is present. To put it mathematically, we denote by aij the
number of offspring of type i produced by a single object of type j.
If we denote by xi(n) the number of the ith type present at time t = n, then the size
of the system at this time is determined by N numbers xl (n) ..... xN(n). Hence, the size
of the i th type at time t - n + 1 will be given by Y~Y-1 aijxj (n). In vector form we have
x (n + 1) = Ax (n), where A = [aij ] and the vector x (n) = (x 1 (n) ..... XN (n)). We are
interested in the asymptotics of x (n) as time n tends to infinity, that is, we are interested in
the behavior of the sequence A n (x (0)), where x(0) is the initial distribution of the system.
The Perron-Frobenius Theorem allows us to solve this problem.
Indeed, as we mentioned after Theorem 32, the sequence {[r(a)]~
1 A nu}, where u is an
arbitrary positive vector, converges to a unique (up to a multiplier) vector that is collinear
with the unit eigenvector v corresponding to the eigenvalue r(A) > 0. This multiplier depends
on our choice of the vector u. In other words, the sequence {x(n)} asymptotically
behaves as [r (A)]n v.

Positive operators 113
Economics: Leontief's model. A very similar application of the Perron-Frobenius Theorem
32 to economics is produced by the Leontief model. Assume that we have N factories
each of which produces a single commodity (or good). To produce each commodity other
commodities must be employed. We denote by aij the amount of the j th good that is
needed to produce one unit of the i th good. These numbers are called the input coefficients
and they are usually assumed to be constant and nonnegative.
Let us denote by di the demand for the i th good and by xi the total output of the i th good
over a fixed period. Then the standard equilibrium equation "Supply -- Demand" yields the
following system of equations:
N
Xi -- Z
j-1
aijxj -- di,
i--1 ..... N,
or (I - A)x -- d. It is clear from the setup that A >~ 0 and also that the off-diagonal entries
of I - A are nonpositive numbers. The economic situation is "feasible" if and only if I - A
is nonsingular and r(A) ~< 1. Under these conditions the system can be uniquely solved for
each output vector d ~> 0 and x - (I - A)-ld ~> 0. For details and various generalizations
see [33,35].
Economics: general equilibrium. Here we shall describe the basic framework of general
equilibrium theory in economics" for details see [23]. The principal mathematical tools
needed in this area are functional analysis and positive functionals rather than operators.
Nevertheless, it seems very appropriate to mention this area in order to demonstrate the
extent to which mathematical sophistication has penetrated work in economics and finance.
The economic intuition of commodities and prices is understood by means of a dual
system (X, X'). The vector space X is the commodity space and X I is the price space. The
evaluation (x, x I) is interpreted as the value of the "bundle" x at prices x ~. For simplicity,
we shall assume here that X is a (finite- or infinite-dimensional) Banach lattice and that X ~
is its norm dual.
An exchange economy consists of a finite number of consumers, say m, indexed by i
(i -- 1 ..... m) whose commodity-price duality is described by (X, X'). Every consumer i
has a preference relation >-i, i.e., a binary relation on X+, which allows the consumer to
distinguish out of any two bundles x and y the one which is better for him. The relation
x ___i Y is read "the bundle x is preferred to y "The preference ___.i is assumed to be complete
(i.e., any two vectors in X+ are comparable), transitive, convex (i.e., the better-than-x set
{y E X+" y ___i X } is a convex set for each x E X+), continuous (i.e., for each x E X+ the
sets {y E X+" y ~i X} and {y E X+" x ___i Y} are both closed), and monotone (i.e., "more
is better" in the sense that y > x ~> 0 implies y ___i X). Each consumer i has also an initial
endowment COi > 0. The total or the social endowment (or the resources) of the economy
is the vector co- Ziml coi.
An allocation (redistribution) is any m-tuple (xl, x2 ..... Xm) such that xi >~ 0 for each i
and Ziml xi - co. The objective of the economic activity in the economy is to redistribute
the resources among the consumers in such a way that each consumer becomes better off.
The efficient ways of redistributing the goods among the consumers are achieved by means

114 Y.A. Abramovich and C.D. Aliprantis
of Pareto optimal allocations. An allocation (x l, X2 ..... Xm) is said to be Pareto optimal
if there is no other allocation (yl, y2 ..... Ym) such that Yi ~-i Xi for each consumer i and
Yi ~i xi for some/(where yi ~i Xi means yi ~i Xi and xi ~i Yi). In other words, a distribution
of the resources is Pareto optimal if it is impossible to make some consumer better
off without making some other one worse off. Another important property of allocations
is that of individual rationality: an allocation (x l, x2 ..... Xm) is individually rational if
Xi ~i (-t)i holds for each consumer i; that is, the redistribution does not give any consumer
a bundle which is worse than his initial endowment. If the order interval [0, co] is weakly
compact (in particular, if X has order continuous norm), then an easy application of Zorn's
lemma guarantees that Pareto optimal individually rational allocations exist.
The above efficiency notion requires that the acceptable redistributions of the goods are
Pareto optimal and individually rational. This brings us to the important decentralization
problem in economics, which in mathematical terminology can be stated as follows: Is
there a price system (i.e., some linear functional x ~ E X ~) that can be used to enforce a
given Pareto efficient allocation? Put another way: Can we use a system of prices - like the
one we see in every day life - to redistribute the resources of the economy?
This problem is related to the mathematical notion of supportability of convex sets by
continuous linear functionals. Recall that a subset A of a vector space is said to be supported
by a non-zero linear functional f at some point a E A if f (a) ~< f (x) holds for all
x 6 A. An allocation (x l, x2 ..... Xm) is said to be supported by a non-zero price x ~ E X ~
if x' supports the better-than-x/set of each consumer i (i.e., the set {x E X+: x ~i Xi }) at
the point xi. That is, x ~ E X~supports the allocation (Xl, x2 ..... Xm) if and only if
x • xi---> (x, x') >>, (xi, x').
Now the economic concept of the decentralization of the allocations finds its best interpretation
in the two famous theorems of welfare economics. These theorems were first proved
for finite dimensional commodity spaces by K. Arrow and G. Debreu and are stated next.
1 ST WELFARE THEOREM. Every allocation supported by prices is Pareto optimal.
2ND WELFARE THEOREM. Every Pareto optimal allocation can be supported by prices.
In particular, the second welfare theorem is of fundamental importance and its validity is
a "must" for any economic system. To establish the validity of this theorem is, in general,
a difficult problem, especially when one deals with an infinite dimensional commodity
space. Supporting a convex set at a given (boundary) point by a non-zero continuous linear
functional is not always feasible and supporting m convex sets each at a specified point by
the same non-zero continuous linear functional is, if not impossible, a very serious matter.
To achieve this goal, we must impose some extra conditions.
The existence of a supporting price for a given Pareto optimal allocation (x 1, x2 ..... Xm)
is proved by utilizing the lattice structure of the price space X ~. It usually consists of two
that supports the
steps. In the first step one proves the existence of a non-zero price x i

Positive operators 115
better-than-x/ set at Xi. Once this is done, then the second step requires a sophisticated
argument to establish that the supremum linear functional
x' -- x' l v x~ v... vx~
is non-zero and supports the allocation (x l, x2 . . . . . Xm). The lattice structure of X' is of
fundamental importance. As a matter of fact, if X' is not a vector lattice, there are examples
of Pareto optimal allocations that cannot be supported by prices.
Finance: choosing the optimal portfolio. We shall discuss here how positive operators
appear in the theory of finance in the context of choosing an optimal portfolio under uncertainty.
For a systematic presentation the reader can consult the excellent book [95].
All investment decisions are made under uncertainty regarding future financial conditions.
To analyze the decision process under uncertainty in the theory of finance, one usually
introduces a two-period model (the simplest possible model), where at period 0 (today)
everything is known and at period 1 (tomorrow) everything is unknown and uncertain. To
capture the notion of uncertainty tomorrow one usually introduces the set S of all possible
states of the world tomorrow. The usual assumptions about S is that it is either a finite set
S -- { 1, 2 ..... S }, or a more general compact Hausdorff topological space, or a probability
space (S, P).
By the term security one should understand any object or property that can be traded
in the markets and that has a value today. While any such object (security) has a specific
value today, its value tomorrow is uncertain and depends, in general, upon the prevailing
state of the world tomorrow. Thus, a security is characterized by a value (or a payoff)
at each possible state of the world tomorrow. Mathematically, this naturally leads to the
interpretation of a security simply as a function x : S --+ R, where x(s) is interpreted as the
value of the security x tomorrow when the state of the world s has been realized. Obvious
economic considerations require that the collection of all securities X is a vector space
under the usual pointwise operations. This space is called the space of securities. The most
common spaces of securities used in finance are C (S) and L2 (S, P); the spaces L p (S, P)
also are used quite often. In sum: the vector space of securities represents every object or
property available that can be possibly traded in the market today.
However, in today's market not every security is available for trade. For instance, every
house is a security, but not everyone's house is in the market for sale today. The collection
M of all securities that are available for trading today is assumed to be a vector subspace
of X, and this space M is called the marketed space. If M -- X, then the markets are called
complete and if M -r X the markets are incomplete. From the practical point of view the
markets are always incomplete. Clearly, the structure of M should play an important role
in making decisions today given the uncertainty of tomorrow. Now let us discuss the hedging
problem. In order to protect their securities, given the uncertainty of tomorrow, people
are forced to buy insurance. An insurance is simply a non-negative marketed security. If
a person owns a security x and wishes the value of this security tomorrow never to fall
below a certain level k (called the striking price or the floor), then he must purchase some
insurance a c M+ = M A X+ so that x + a ~> k, or x(s) + a(s) >7 k for each state s of the
world tomorrow. This implies a ~ k - x, and hence a ~> (k - x) +, where (k - x)+(s) =

116 YA. Abramovich and C.D. Aliprantis
max{k - x(s), 0}. Therefore, the "smallest" desired insurance seems to be (k - x) + and it
is called the put option of the security x with striking price k. Since M need not be a vector
sublattice of X, the put option (k - x) + need not be available in the market today, i.e.,
(k - x) + need not belong to M. This is why the theory of lattice-subspaces plays an important
role here along with the theory of Banach and vector lattices (see for instance [ 14,24]).
And now we can state the hedging problem of the theory of finance.
THE HEDGING PROBLEM. Given a marketed security x, what is the "cheapest insurance"
that one can purchase in order to guarantee at least the value k for the security
tomorrow no matter what the prevailing state will be?
Of course, if M is a vector sublattice of X and x, k 6 M, then a = (k - x) + should be the
desired answer. The general solution of the Hedging Problem is non-trivial and is related
to the meaning of the "cheapest insurance"9 This is the place where positive operators and
their properties make their appearance.
To simplify matters further, let us assume that there is either a finite or a countable
number of linearly independent securities whose linear span is M. Considering the simplest
case of the two, we will suppose that there are N securities x l, x2 ..... XN in M+ that span
M and that S = { 1, 2 ..... S}. In this case we have N O}.

Positive operators 117
Then a price p 6 M* is arbitrage free if and only if (p, RO) = (R* p, 0) > 0 for each 0 6 K,
where R* :M* --+ ~I~ u is the adjoint operator of R. This duality property is very useful in
determining the arbitrage free prices. From economic considerations only arbitrage free
prices are acceptable prices and the Hedging Problem can be stated now as follows: Given
an arbitrage free price p 6 M*, a security x and a striking price k, find the portfolio 0 E R x
that minimizes the expression (R* p, O) = (p, RO) subject to RO ~ x x/ k.
The Hedging Problem is also related to the notion of a derivative in the theory of finance
and has been studied in many contexts. A derivative is a security of the form
x(s) = f(yl(s), y2(s) ..... yk(s)), where (in the case X = C(S)) f :R k --+ R is a continuous
function and yl, y2 ..... yk are k given securities; that is a derivative is a security
whose payoff depends on the payoffs of a finite number of other securities.
Acknowledgment
The authors would like to express their thanks to A. Kitover and V. Troitsky for many
useful suggestions.
Added in Proof
Detailed proofs of the results discussed in this survey as well as an extensive treatment of
many related topics can be found in the forthcoming book "An invitation to Operator Theory"
by the authors that will be published shortly by the American Mathematical Society.
References
[1] Y.A. Abramovich, Weakly compact sets in topological Dedekind complete vector lattices, Teor. Funkcii
Funkcional. Anal. i Prilo~en. 15 (1972), 27-35.
[2] Y.A. Abramovich, On spaces of operators acting between Banach lattices, Zap. Nauchn. Sem. Leningrad.
Otdel. Mat. Inst. Steklov. (LOMI), Investigations on Linear Operators and Theory of Functions VIII 73
(1977), 182-193.
[3] Y.A. Abramovich, When each continuous operator is regular, Funct. Analysis, Optimization and Math.
Economics, Oxford Univ. Press (1990), 133-140.
[4] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Positive operators on Krein spaces, Acta Appl.
Math. 27 (1992), 1-22.
[5] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, On the spectral radius of positive operators, Math.
Z. 211 (1992), 593-607. Corrigendum: Math. Z. 215 (1994), 167-168.
[6] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Invariant subspaces of operators on e p-spaces,
J. Funct. Anal. 115 (1993), 418-424.
[7] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Invariant subspaces ofpositive operators, J. Funct.
Anal. 124 (1994), 95-111.
[8] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Invariant subspaces for positive operators acting
on a Banach space with basis, Proc. Amer. Math. Soc. 123 (1995), 1773-1777.
[9] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Local quasinilpotence, cycles and invariant subspaces,
Proceedings of the Conference Interaction Between Functional Analysis, Harmonic Analysis, and
Probability, Lecture Notes Pure Appl. Math. 175, N. Kalton, E. Saab and S. Montgomery-Smith, eds,
Marcel Dekker, New York (1995), 1-12.

118 YA. Abramovich and C.D. Aliprantis
[10] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Another characterization of the invariant subspace
problem, Operator Theory in Function Spaces and Banach Lattices, Operator Theory Advances and Applications
75, C.B. Huijsmans, M.A. Kaashoek, W.A.J. Luxemburg and B. de Pagter, eds, Birkh~iuser, Boston
(1995), 15-31.
[11] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Multiplication and compact-friendly operators,
Positivity 1 (1997), 171-180.
[12] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, The invariant subspace problem: some recent advances,
Rend. Inst. Mat. Univ. Trieste 29 (1998), 1-76.
[13] Y.A. Abramovich, C.D. Aliprantis, O. Burkinshaw and A.W. Wickstead, A characterization of compactfriendly
multiplication operators, Indag. Math. 8 (1998), 1-11.
[14] Y.A. Abramovich, C.D. Aliprantis and I.A. Polyrakis, Lattice-subspaces and positive projections, Proc.
Royal Irish Acad. 94A (1994), 237-253.
[15] Y.A. Abramovich, E.L. Arenson and A.K. Kitover, Banach C(K)-modules and Operators Preserving
Disjoinmess, Pitman Research Notes in Mathematical Series 277, Longman Scientific & Technical (1992).
[ 16] Y.A. Abramovich and V.A. Geyler, On a question of Fremlin concerning order bounded and regular operators,
Colloq. Math. 46 (1982), 15-17.
[17] Y.A. Abramovich and L.E Janovsky, Applications of the Rademacher systems to operator characterizations
of Banach lattices, Coll. Math. 46 (1982), 75-78.
[ 18] Y.A. Abramovich and A.W. Wickstead, Regular operators from and into a small Riesz space, Indag. Math.
2 (1991), 257-274.
[19] Y.A. Abramovich and A.W. Wickstead, The regularity of order bounded operators, into C(K), II, Quart.
J. Math. Oxford 44 (1993), 257-270.
[20] Y.A. Abramovich and A.W. Wickstead, Recent results on the order structure of compact operators, Bull.
Irish Math. Soc. 32 (1994), 32-45.
[21] Y.A. Abramovich and A.W. Wickstead, When each continuous operator is regular, II, Indag. Math. 8
(1997), 281-294.
[22] C.D. Aliprantis and K.C. Border, Infinite Dimensional Analysis, 2nd edn, Springer, Berlin (1998).
[23] C.D. Aliprantis, D.J. Brown and O. Burkinshaw, Existence and Optimality of Competitive Equilibria,
Springer, New York (1990).
[24] C.D. Aliprantis, D.J. Brown and J. Wemer, Minimum-cost portfolio insurance, J. Econom. Dynamics
Control 24 (2000), 1703-1719.
[25] C.D. Aliprantis and O. Burkinshaw, Positive compact operators on Banach lattices, Math. Z. 174 (1980),
289-298.
[26] C.D. Aliprantis and O. Burkinshaw, On weakly compact operators on Banach lattices, Proc. Amer. Math.
Soc. 83 (1981), 573-578.
[27] C.D. Aliprantis and O. Burkinshaw, Dundord-Pettis operators on Banach lattices, Trans. Amer. Math.
Soc. 274 (1982), 227-238.
[28] C.D. Aliprantis and O. Burkinshaw, Factoring compact and weakly compact operators through reflexive
Banach lattices, Trans. Amer. Math. Soc. 183 (1984), 369-381.
[29] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York (1985).
[30] I. Amemiya, A generalization of Riesz-Fischer's theorem, J. Math. Soc. Japan 5 (1953), 353-354.
[31] T. And& Positive operators in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ., Ser. 1 13 (1957),
214-228.
[32] I.A. Bakhtin, M.A. Krasnoselsky and V.Ya. Stezenko, Continuity of linear positive operators, Sibirsk.
Math. Z. 3 (1962), 156-160.
[33] R.B. Bapat and T.E.S. Raghavan, Nonnegative Matrices and Applications, Cambridge Univ. Press, Cambridge
(1997).
[34] A. Benedek and R. Panzone, The spaces with mixed norm, Duke Math. J. 28 (1961), 301-324.
[35] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press,
New York (1979).
[36] L. de Branges, A construction ofinvariant subspaces, Math. Nachr. 163 (1993), 163-175.
[37] A.V. Bukhvalov, On integral representation of linear operators, Zap. Nauchn. Sem. Leningrad Otdel. Mat.
Inst. Steklov. (LOMI) 47 (1974), 5-14.
[38] A.V. Bukhvalov, Nonlinear majorization of linear operators, Soviet Math. Dokl. 37 (1988), 4-7.

Positive operators 119
[39] A.V. Bukhvalov, A.I. Veksler and G.Ya. Lozanovsky, Banach lattices - some Banach aspects of the theory,
Russian Math. Surveys 34 (1979), 159-212.
[40] A.V. Bukhvalov, A.I. Veksler and V.A. Geyler, Normed lattices, Mathematical Analysis 18, Akad. Nauk
SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow (1980), 125-184.
[41] A.V. Bukhvalov, A.E. Gutman, V.B. Korotkov, A.G. Kusraev, S.S. Kutateladze and B.M. Makarov, Vector
Lattices and Integral Operators, Mathematics and its Applications 358, Kluwer, Dordrecht (1996).
(Translated from the 1992 Russian original.)
[42] R. Btirger, Mutation-selection balance and continuum-of-alleles models, Math. Biosci. 91 (1988), 67-83.
[43] R. Btirger and I.M. Bomze, Stationary distributions under mutation-selection balance: structure and properties,
Adv. Appl. Probab. 28 (1996), 227-251.
[44] D.I. Cartwright and H.E Lotz, Some characterizations of AM- and AL-spaces, Math. Z. 142 (1975), 97-
103.
[45] V. Caselles, On irreducible operators on Banach lattices, Indag. Math. 48 (1986), 11-16.
[46] V. Caselles, On band irreducible operators on Banach lattices, Quaestiones Math. 10 (1987), 339-350.
[47] Z.L. Chen, The factorization of a class of compact operators, Dongbei Shuxue 6 (1990), 303-309.
[48] M.D. Choi, E.A. Nordgren, H. Radjavi, H.P. Rosenthal and Y. Zhong, Triangularizing semigroups of
quasinilpotent operators with non-negative entries, Indiana Univ. Math. J. 42 (1993), 15-25.
[49] E.N. Dancer, Positivity of maps and applications. Topological nonlinear analysis, Progr. Nonlinear Differential
Equations Appl. 15, Birkh~iuser, Boston (1995), 303-340.
[50] W.J. Davis, T. Figiel, W.B. Johnson and A. Petczyfiski, Factoring weakly compact operators, J. Funct.
Anal. 17 (1974), 311-327.
[51] J. Diestel and J.J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, RI (1977).
[52] P.G. Dodds and D.H. Fremlin, Compact operators in Banach lattices, Israel J. Math. 34 (1979), 287-320.
[53] R. Drnov~ek, Triangularizing semigroups of positive operators on an atomic normed Riesz space, Proc.
Edinburgh. Math. Soc. 43 (2000), 43-55.
[54] N. Dunford and J.T. Schwartz, Linear Operators, I, Wiley (Interscience), New York (1958).
[55] N. Dunford, Integration and linear operations, Trans. Amer. Math. Soc. 40 (1936), 474-494.
[56] E Enflo, On the invariant subspace problem for Banach spaces, Seminaire Maurey-Schwarz (1975-1976);
Acta Math. 158 (1987), 213-313.
[57] T. Figiel, Factorization of compact operators and applications to the approximation property, Studia Math.
45 (1973), 191-210.
[58] D.H. Fremlin, Topological Riesz Spaces and Measure Theory, Cambridge Univ. Press, London (1974).
[59] D.H. Fremlin, A characterization of L-spaces, Indag. Math. 36 (1974), 270-275.
[60] G. Frobenius, Uber Matrizen aus nicht-negativen Elementen, Sitzungsber. Kgl. Preul3. Akad. Wiss. Berlin
(1912) 456-477.
[61] N. Ghoussoub, Positive embeddings of C(A), L!, g-1 (1-') and (~-~n @g'~)~l Math. Ann. 262 (1983), 461-
472.
[62] N. Ghoussoub and W.B. Johnson, Counterexamples to several problems on the factorization of bounded
linear operators, Proc. Amer. Math. Soc. 92 (1983), 461-472.
[63] N. Ghoussoub and W.B. Johnson, Factoring operators through Banach lattices not containing C(O, 1),
Math. Z. 194 (1987), 194-171.
[64] G. Godefroy, Sur les op~rateurs compacts r~guliers, Bull. Soc. Belg. A 30 (1978), 35-37.
[65] J.J. Grobler, A short proofofthe Andf-Krieger theorem, Math. Z. 174 (1980), 61-62.
[66] J.J. Grobler, Band irreducible operators, Indag. Math. 48 (1986), 405-409.
[67] P.R. Halmos and V.S. Sunder, Bounded Integral Operators on L 2 Spaces, Springer, Berlin (1978).
[68] C.B. Huijsmans and W.A.J. Luxemburg, eds, Positive Operators and Semigroups on Banach Lattices,
Kluwer, Dordrecht (1992).
[69] M.T. Jahandideh, On the ideal-triangularizability of positive operators on Banach lattices, Proc. Amer.
Math. Soc. 125 (1997), 2661-2670.
[70] G. Jameson, Ordered Linear Spaces, Lecture Notes in Math. 141, Springer, Heidelberg (1970).
[71] W.B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345.
[72] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the
Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001),
1-84.

120 Y.A. Abramovich and C.D. Aliprantis
[73] N.J. Kalton and E Saab, Ideal properties of regular operators between Banach lattices, Illinois J. Math.
29 (1985), 382-400.
[74] L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon, New York (1982).
[75] L.V. Kantorovich, B.Z. Vulikh and A.G. Pinsker, Functional Analysis in Partially Ordered Spaces,
Gostekhizdat, Moscow (1950).
[76] L.V. Kantorovich and B.Z. Vulikh, Sur la representation des op&ations lin~ares, Compositio Math. 5
(1937), 119-165.
[77] A.K. Kitover, The spectral properties of weighted homomorphisms in algebras of continuous functions and
their applications, Zap. Naurn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 107 (1982), 89-103.
[78] V.B. Korotkov, Integral Operators, Nauka, Novosibirsk (1983).
[79] M.A. Krasnoselsky, P.P. Zabre~o, E.I. Pustylnik and P.E. Sobolevsky, Integral Operators in Spaces of
Summable Functions, Noordhoff International Publishing, Leiden (1976).
[80] M.A. Krasnoselsky, E.A. Lifshits and A.V. Sobolev, Positive Linear Systems. The Method of Positive
Operators, Sigma Series in Applied Mathematics 5, Heldermann, Berlin (1989).
[81] M.G. Krein, Fundamental properties of normal conical sets in a Banach space, Dokl. Akad. Nauk USSR
28 (1940), 13-17.
[82] M.G. Krein and M.A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi
Mat. Nauk 3 (1948), 3-95. (Russian). Also, Amer. Math. Soc. Transl., Vol. 26 (1950).
[83] S.G. Krein, Yu.I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Trans. Math. Monogr.
54, Amer. Math. Soc., Providence, RI (1982).
[84] U. Krengel, Ergodic Theorems, de Gruyter Studies in Math. 6, Walter de Gruyter, Berlin (1985).
[85] H.J. Krieger, Beitrfige zur Theorie positiver Operatoren, Schrifienreihe der Institute fiir Math. Reihe ,4,
Heft 6, Akademie-Verlag, Berlin (1969).
[86] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, Berlin (1977).
[87] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces H, Springer, Berlin (1979).
[88] V.I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous
operator, Funktsional. Anal. i Prilozhen 7 (3) (1973), 55-56.
[89] V.I. Lomonosov, An extension of Burnside's theorem to infinite-dimensional spaces, Israel J. Math. 75
(1991), 329-339.
[90] G.Ya. Lozanovsky, On almost integral operators on KB-spaces, Vestnik Leningr. Univ. Mat. Meh. Astronom.
7 (1966), 35-44.
[91] G.Ya. Lozanovsky, On a theorem ofN. Dunford, Izvestiya VUZ'ov, Mathematika 8 (1974), 58-59.
[92] W.A.J. Luxemburg, Notes on Banach function spaces, Indag. Math. 27 (1965), Note XIV, 229-248; Note
XV, 415-446; Note XVI, 646-667.
[93] W.A.J. Luxemburg and A.C. Zaanen, Notes on Banach function spaces, Indag. Math. 25 (1963), Note
I, 135-147; Note II, 148-153; Note III, 239-250; Note IV, 251-263; Note V, 496-504; Note VI, 655-
668; Note VII, 669-681; also Indag. Math. 26 (1964), Note VIII, 104-119; Note IX, 360-376; Note X,
493-506; Note XI, 507-518; Note XII, 519-529; Note XIII, 530-543.
[94] W.A.J. Luxemburg and A.C. Zaanen, Riesz Spaces I, North Holland, Amsterdam (1971).
[95] M. Magill and M. Quinzii, Theory oflncomplete Markets, MIT Press, Cambridge, MA (1996).
[96] P. Meyer-Nieberg, Banach Lattices, Springer, Berlin (1991).
[97] A.J. Michaels, Hilden's simple proof of Lomonosov's invariant subspace theorem, Adv. in Math. 25 (1977),
56-58.
[98] H. Minc, Nonnegative Matrices, Wiley, New York (1988).
[99] L. Nachbin, On the continuity of positive linear transformations, Proc. Internat. Congress of Math. 1
(1950), 464--465.
[100] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo (1950).
[ 101 ] I. Namioka, Partially ordered linear topological spaces, Mem. Amer. Math. Soc. 24 (1957).
[ 102] R.D. Nussbaum, Eigenvectors ofnonlinear positive operators and the linear Krein-Rutman theorem, Fixed
Point Theory, Lecture Notes in Math. 886, E. Fadell and G. Fourier, eds, Springer, Berlin (1981), 309-330.
[ 103] R.D. Nussbaum, The Fixed Point Index and Some Applications, Srminaire de Mathrmatiques Suprrieures
(Seminar on Higher Mathematics) 94, Presses de l'Universit6 de Montrral, Montreal, Quebec (1985).

Positive operators 121
[104] R.D. Nussbaum, The fixed point index and fixed point theorems, Topological Methods for Ordinary Differential
Equations (Montecatini Terme, 1991), Lecture Notes in Math. 1537, Springer, Berlin (1993),
143-205.
[105] P. Orno, On Banach lattices of operators, Israel J. Math. 19 (1974), 164-165.
[106] B. de Pagter, Irreducible compact operators, Math. Z. 192 (1986), 149-153.
[107] A.L. Peressini, Ordered Topological Vector Spaces, Harper & Row, New York (1967).
[108] O. Perron, Zur Theorie der Matrizen, Math. Ann. 64 (1907), 248-263.
[109] L.D. Pitt, A compacmess condition for linear operators on function spaces, J. Oper. Theory 1 (1979),
49-54.
[110] I.M. Popovici and D.T. Vuza, Factoring compact operators and approximable operators, Zeit. Anal.
Anwendungen 9 (1990), 221-233.
[111] I.M. Popovici and D.T. Vuza, Factorization of convolution operators, Analele Univ. Craiova 16 (1988),
1-8.
[ 112] H. Radjavi, The Perron-Frobenius theorem revisited, Positivity 3 (1999), 317-331.
[113] A. Rhandi, Positivity and stability for a population equation with diffusion on L 1, Positivity 2 (1998),
101-113.
[114] A. Rhandi and R. Schnaubelt, Asymptotic behavior of a non-autonomous population equation with diffusion
in L 1, Preprint (1998).
[115] C.J. Read, A solution to the invariant subspace problem on the space e ! , Bull. London Math. Soc. 17
(1985), 305-317.
[116] C.J. Read, A short proof concerning the invariant subspace problem, J. London Math. Soc. 34 (1986),
335-348.
[ 117] C.J. Read, Quasinilpotent operators and the invariant subspace problem, J. London Math. Soc. 56 (1997),
595-606.
[118] A.C.M. van Rooij, On the space of all regular operators between two Riesz spaces, Indag. Math. 47 (1985),
95-98.
[119] A.C.M. van Rooij, When do the regular operators between two Riesz spaces form a Riesz space?, Report
No. 8410, Dept. of Math., Catholic University Nijmegen (1984).
[120] H.H. Schaefer, Topologische Nilpotenz irreduzibler Operatoren, Math. Z. 117 (1970), 135-140.
[ 121 ] H.H. Schaefer, Topological Vector Spaces, Springer, Berlin (1971).
[122] H.H. Schaefer, Normed tensor products of Banach lattices, Israel J. Math. 13 (1972), 400-415.
[123] H.H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin (1974).
[124] H.H. Schaefer, On theorems ofde Pagter andAndO-Krieger, Math. Z. 192 (1986), 155-157.
[125] A.R. Schep, Kernel operators, Indag. Math. 41 (1979), 39-53.
[126] H.U. Schwarz, Banach Lattices and Operators, Teubner Texte 71, Leipzig (1984).
[127] A. Simoni~, A construction of Lomonosov functions and applications to the invariant subspace problem,
Pacific J. Math. 175 (1996), 257-270.
[128] A. Simoni6, An extension of Lomonosov's techniques to non-compact operators, Trans. Amer. Math. Soc.
348 (1995), 975-995.
[ 129] J. Synnatzschke, On the conjugate of a regular operator and some applications to the questions of complete
continuity and weak complete continuity of regular operators, Vestnik Leningr. Univ. Mat. Meh. Astronom.
1 (1972), 60-69.
[130] M. Talagrand, Some weakly compact operators between Banach lattices do not factor through reflexive
Banach lattices, Proc. Amer. Math. Soc. 96 (1986), 95-102.
[131] V.G. Troitsky, On the modulus of C.J. Read's operator, Positivity 2 (1998), 257-264.
[132] V.G. Troitsky, Lomonosov's theorem cannot be extended to chains of four operators, Proc. Amer. Math.
Soc. 128 (2000), 527-540.
[133] L. Tzafriri, On Banach spaces with unconditional bases, Israel J. Math. 17 (1974), 84-93.
[134] B.Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Wolters-Noordhoff, Groningen
(1967).
[ 135] B.Z. Vulikh, Introduction to the Theory of Cones in Normed Spaces, Kalinin State University (1977).
[136] B.Z. Vulikh, Special Topics on Geometry of Cones in Normed Spaces, Kalinin State University (1978).
[137] B. Walsh, On characterizing K6the sequence spaces as vector lattices, Math. Ann. 175 (1968), 253-256.
[138] A.W. Wickstead, Compact subsets of partially ordered Banach spaces, Math. Ann. 212 (1975), 271-284.

122 Y.A. Abramovich and C.D. Aliprantis
[139] A.W. Wickstead, Extremal structure of cones of operators, Quart. J. Math. Oxford Ser. (2) 32 (1981),
239-253.
[140] A.W. Wickstead, The regularity oforder bounded operators into C(K), Quart. J. Math. Oxford 44 (1993),
257-270.
[ 141 ] A.W. Wickstead, Dedekind completeness of some lattices of compact operators, Bull. Polon. Acad. Sci.
43 (1995), 297-304.
[142] A.W. Wickstead, Converses for the Dodds-Fremlin and Kalton-Saab theorems, Math. Proc. Camb. Phil.
Soc. 120 (1996), 175-179.
[143] Y.C. Wong and K.E Ng, Partially Ordered Topological Vector Spaces, Clarendon Press, Oxford (1973).
[144] A.C. Zaanen, Riesz Spaces H, North-Holland, Amsterdam (1983).
[145] A.C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer, Berlin (1997).
[ 146] P.E Zabre~o and S.V. Smickih, A theorem ofM. G. Kre~n and M.A. Rutman, Funktsional. Anal. i Prilozhen.
13 (3) (1979), 81-82.

CHAPTER 3
Lp Spaces
Dale Alspach
Department of Mathematics, Oklahoma State University, Stillwater, OK, USA
E-mail: alspach @ math. okstate, edu
Edward Odell*
Department of Mathematics, The University of Texas, Austin, TX, USA
E-mail: odell@ math. utexas, edu
Contents
1. Preliminaries .................................................. 126
2. Global structure ................................................. 129
3. Sequences in ~ p and L p ............................................ 134
4. Subspaces of L p ................................................ 140
5. /2p-spaces, 1 < p < cx~, p-f: 2 ......................................... 146
References ..................................................... 156
*Research supported by NSE
HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 1
Edited by William B. Johnson and Joram Lindenstrauss
9 2001 Elsevier Science B.V. All rights reserved
123

L p spaces 125
In this chapter we will discuss the structure of the L p-spaces and their subspaces.
We are concerned mainly with the reflexive case (1 < p < ~). The space L~ and the
/21-spaces are considered elsewhere [90]. A theory requires and feeds on its examples. The
L p-spaces have provided much fodder for the general theory of Banach spaces because
they appeared early in the theory and the study of these spaces has motivated the definitions
of many properties of more general Banach spaces. For example, with its usual norm L p
is a Banach lattice under the pointwise almost everywhere ordering. In the reflexive case it
also has an unconditional basis and thus has another lattice structure as a sequence space
(and is a Banach lattice under a different norm except in the case p - 2). These spaces
naturally occur as interpolation spaces and are the simplest of the rearrangement invariant
spaces. The Hardy spaces, Hp(D), of analytic functions on the unit disk with Lp boundary
values are isomorphic to Lp, 1 < p < cxz, and the Bergman spaces Ap(D) of analytic
functions on the unit disk which are in Lp(D) are isomorphic to ~p, 1 ~< p < e~, [67]. The
study of the structure of the finite dimensional subspaces of L p paved the way for much
of the extraordinary development of the local theory of Banach spaces in the 1980's [77].
In investigations of other Banach spaces and operators the existence and classification of
operators from, into or factoring through L p-spaces provide fundamental information on
the structure.
We will concern ourselves primarily with the infinite dimensional isomorphic structure
of the Lp-spaces. In particular we shall concentrate on separable Lp-spaces and, as noted
in the basic concepts chapter, [49, Section 4], this reduces essentially to studying g p and
L p(O, 1) (which we shall denote by L p). Functions are assumed to be real-valued as in
general there are only minor adjustments needed for the complex case.
As the theory of Lp-spaces was developed some natural questions arose. What are the
complemented subspaces of Lp (or ~p).9 When does a given Banach space X embed into
Lp (or gp)? If X c_ Lp, what subspaces must live inside of X? Does every Banach space
contain some g p or co? What can be said about the structure of an unconditional basic
sequence in L p ? We address these problems and others below. With apologies to the experts
we choose to include some of the well known (to them) structural results that were
mentioned in the basic concepts chapter or appear in books such as [70].
In Section 1 we review certain inequalities for sequences in L p. For example we show
that by integrating against the Rademacher functions one can deduce that a normalized
unconditional basic sequence in Lp (2 < p < CX~) admits upper ~2 and lower gp estimates
(respectively, for 1 < p < 2, upper gp and lower ~2 estimates). In Section 2 we study the
global structure of L p and in particular the Haar basis. Among the results given we examine
the span of subsequences of the Haar basis and Schechtman's result that every complemented
subspace of L p with an unconditional basis is isomorphic to a complemented
subspace spanned by a block basis of the Haar basis.
Section 3 begins with some properties and characterizations of the unit vector basis of
p. The details of the argument using the Pelczyfiski decomposition method that a complemented
subspace of g p is isomorphic to g p are presented to help give context and meaning
to the notion of (p, 2) bounded operators discussed later in Section 5. We also discuss
spreading models and types and the Krivine-Maurey theorem. This latter result gives a
sufficient condition, in terms of types, for a Banach space X to contain almost isometric
copies of g p for some p. Section 4 deals with subspaces X of L p. For example if

126 D. Alspach and E. Odell
X C Lp (2 < p < cx~) then X must contain an isomorph of s or s and if X does not
contain s it must embed into s We also consider a necessary and sufficient condition for
a reflexive space X to embed into the s sum (1 < p < cx~) of finite dimensional spaces.
The last section concerns/2p-spaces- those complemented subspaces of L p which are not
Hilbert spaces. Here the known examples and their isomorphic classification are discussed.
In particular Rosenthal's space X p and its generalizations are presented in detail. The role
of (p, 2)-bounded operators is also explained. Finally we discuss some of the results for
s
with )~ near one and their relation to the isometric theory of Lp-spaces.
1. Preliminaries
We first recall a few key properties of L p and s
concepts chapter.
The unit vector basis for s
which are discussed throughout the basic
is a 1-symmetric basis [49, Section 3]. The Haar basis
(hi)~ is an unconditional basis of Lp for 1 < p < c~ [49, Section 3], [24]. It is also a
monotone basis for Lp for 1 ~< p < c~. The Rademacher functions (rn)n~=l , [49, Section
4], are equivalent to the unit vector basis of s for p < ~, (and the unit vector basis of el
for p=~).
Thus for 0 < p < oc there exist constants A p, B p with
Ap(~lanl2)l/2 (fo I )l/p 2)1/2

L p spaces 127
if 1 ~< p ~< 2, and
(i011
Ilxillp ~< ~ri(t)xi
1
dt

128 D. Alspach and E. Odell
(1.2) and (1.3) can be viewed as generalizations of Clarkson's inequalities [29]. Since
II 9 IILp ~< I1" IlL2 for p ~< 2 we also have using (1.2) for p = 2 that
(So ~ri(t)xi
1
dt ~< IIx/ll 2
p
1/2
for 1 ~ p < 2, (1.5)
and similarly
(So
~-~ ri (t)xi
1
at 1> ilxill 2
p
1/2
for 2 < p. (1.6)
The technique of integrating against the Rademacher yields some useful inequalities for
unconditional basic sequences in L p. If (Xn) is a A-unconditional basic sequence in L p
then
[f01 2)p/2 ]l/p
A-I (Elanl2iXn(S)l as I[Ea, x, II"

independent mean zero random variables in L p then
L p spaces 129
max Ilxi IlPp , Ilxi II 2
i:1 i--1
~Xi
t
~ Kpmax Ilxi lip
i:1 p i=1
, IIx/ll 9 (1.11)
It is shown in [55] that Kp ~ p~ In p.
A Banach space X is a L;p,Z-space if for all finite dimensional spaces F ___ X there exists
a finite dimensional E with F c E c X so that d(E fdimE) ~ )~ It ultimately turns out
m ~ vp ,
(see Section 5) that a separable X is s for some )~ and 1 < p < ec iff X is isomorphic
to a complemented subspace of L p which is not isomorphic to Hilbert space [66,68].
The situation for L 1 is more complicated. It is conjectured that every infinite dimensional
complemented subspace X of L1 is isomorphic to L1 or el. It is known that if X contains
an isomorph of L1 then X is isomorphic to L1 [36] and if X embeds into el then X is
isomorphic to el [65]. Various characterizations of/21- (and L;oc-) spaces are given in [68].
Much work was done to study and attempt to classify the L;p-spaces up to isomorphism
and this is discussed in Section 5 below.
We begin with some results on the global structure of L p and in particular those involving
the Haar basis. All Banach spaces are presumed to be separable unless otherwise
stated.
2. Global structure
In Hilbert space strong geometric properties follow from the properties of the inner product.
Early in the development of operator theory on more general spaces Murray [81 ] recognized
the critical role that the orthogonal projection played for operators on L2 and
attempted to find generalizations for the L p-spaces. While we know now (and as Murray
himself discovered in part) that there are far fewer bounded projections on the typical Banach
spaces than on Hilbert space, nevertheless, spaces such as L p possess a rich family
of projections and of corresponding complemented subspaces.
Given the unconditionality of the Haar basis the simplest way to get a complemented
subspace of Lp (1 < p < oc) is to take [(hji)]~__l where (hji) is a subsequence of (hj).
There are however only two isomorphism types of such subspaces.
THEOREM 1 ([39]). Let 1 < p < oc and let X -- [(h j;)] for some subsequence (hji) of
(h j). Then X is isomorphic to e p or L p.
PROOF. By the decomposition method [49, Section 4], (or see the proof of Theorem 15
c c c
below) it suffices to prove that either X ~ e p or L p ~ X. (X ~ Y means that X is
isomorphic to a complemented subspace of Y.) Let
A = { t c [0, 1 ]" t c supp hji for infinitely many i's}.

- {t
- U(An
130 D. Alspach and E. Odell
C
We shall show that X ~ s if m (A) -- 0 and otherwise L p ~ X.
So assume m(A) = 0. We shall produce a sequence (Sn) of disjoint measurable subsets
of [0, 1] with m ([0, 1] \ Un Sn) = 0 and such that every element of X is constant on each
Sn. It then follows that
C
X ~-+ [(l&)]~ g.p.
To do this it suffices to show that if n E N and
An -
~ [0, 1]" t ~ supphji for exactly n i's},
then An may be split into such a sequence of sets. Let (Xi) be the subsequence of those
h ji's involved in the definition of An and let (yi) be the (possibly finite) subsequence of
(xi) determined by Ui supp yi ~ An and if Xn :/= Yi then suppxn g supp Yi. The yi's are
disjointly supported and
An -
n supp y+) U U(An n supp y?).
i
i
This gives the desired splitting of An.
If m(A) > 0 we partition (hji) into a countable number of subsequences (hji)iEN k so
that supp hji A supp h je = 0 if i -J= e 6 Ark and so that if k >~ 2, i ~ Nk then there exists
~ Nk-1 with supphji c_ supphje. Set Bk = UicNk supphji" Clearly
B1 ___ B2 ___ "" and (-]Bk -- A.
One now chooses a subsequence (kn) of N with m (Bk, \ A) < en, en .[. 0 rapidly. Define
do- L
and inductively for m odd
d,- L hji
icNk o i~Nk 1
d2"+m -- Z
hji summed over {i 6 Ark," supphji ___ suppd2n_l+~(m+l)/2]]
+ },
and for m even
d2n+m-Zhji summed over {i ~ Nk," supphji c_ d~_,+~(m+,)/2~}.
A perturbation argument yields that (dn) is equivalent to the Haar basis and it is not hard
to see that [(dn)] is complemented in X, [10].
D
A vector-valued version of this theorem exists [78]. There also exists a finite dimensional
version of this theorem.

L p spaces 131
THEOREM 2 ([80]). Let 1 < p < cx~. There exists Kp < ~X) so that if (hji)in___l is anyfinite
subsequence of (hj) and E -- ((hji)~) then d(E, ~np) 1.)
(b) ([79]) Let 1 < p :/= 2 < cx~. There exists Kp >/ 1 and for all n a rearrangement
(hn~2 n 2 n n 2 n
~'i 'i=0 ~ (hi)i=o so that if M E N and (zi) M is a normalized block basis of (b i )i=0 then
there exists A c {1 ..... M} with [A[ ~> M/2 so that (Zi)iEA is Kp equivalent to the unit
Ial
vector basis of g~ p .
The rearrangement is defined in terms of the supports of the Haar functions. For i < j
either supp b n and supp b n j are disjointwithsuppbi n lying to the left of supp bj n orsuppbi n _
suppbj.
Every s has a basis ([54], see Section 5 below) but it is still open as to whether
or not it must have an unconditional basis. However those that do can be realized isomorphically
as being spanned by block bases of (hn).
THEOREM 4 ([95]). Let 1 < p < cx~ and let X be a complemented subspace of Lp with an
unconditional basis (Xn). Then there exists a block basis (Yn) of the Haar basis with [(Yn)]
complemented in L p and such that (Xn) is equivalent to (Yn).
The first part of this theorem follows from the following proposition.
PROPOSITION 5 ([95]). Let (Xn) be an unconditional basic sequence in Lp (1 < p < ~).
Then there exists an equivalent block basis (Yn) of (hn).
PROOF. From (1.7) and (1.8) it follows that if (yi) is another unconditional basic sequence
in Lp with [yi[ -- Ixi[ then (yi) is equivalent to (xi). By a perturbation argument we may
assume that there exist integers m n ~ ~ and scalars (ai) so that
2 mn - l
Xn-- Z azmn+klh2mn+k["
k=0
(2.1)
One defines the yn'S via this formula by removing the absolute values. [3

fi2
132 D. Alspach and E. Odell
Before completing the proof of Theorem 4 we recall a useful isomorphic representation
of Lp as Lp(~2) for 1 < p < cxz. The latter is the Banach space of all sequences (fi) of
measurable functions on [0, 1] with
II(fi)llL,r -
(t) dt 2 it can be shown that (fij) is unconditional and since (fi) is
a subsequence of (fij) we obtain a projection of Lp(s onto [(fi)]. (fi) is equivalent
to (xi) and is a block basis of (h 2mn+e) ~'2mn--1 1,k=0 Furthermore (h 2mn+k) is equivalent to
n~
(h2mn+k) and this completes the prooffor p > 2. For p < 2 a duality argument is necessary
[951. D
Some Banach spaces have the property that whenever they are isomorphic to a subspace
of another nice enough Banach space that there is another isomorphic embedding with better
properties. One property of this type is reproducibility. A basis (xi) for a Banach space
X is reproducible if whenever X ___ Y and Y has a basis (yi) then some block basis of (yi)
is equivalent to (xi). It is easy to see that because they are weakly null and subsymmetric
that the unit vector bases of s (1 < p < cxz) or co are reproducible. It is also not hard to
show that the unit vector basis of s is reproducible.
THEOREM 6 ([67]). The Haar basis for Lp (1 ~< p < cx~) is a reproducible basis. Moreover
the same is true for any unconditional basis of Lp (1 < p < cx~).
PROOF. By virtue of Proposition 5 it suffices to prove that (hi) is a reproducible basis for
L p for 1 ~< p < oc. Let L p c X where X has a basis (Xn) with biorthogonal functionals
(Xn*). By Liapounoff's convexity theorem for each finite set F ___ X* and A ___ [0, 1] with

Lp spaces 133
m(A) > 0 there exists a measurable set B c_ A with m(B) -- m(A \ B) and x*(1B) --
x*(1A\B) for x* E F.
Let en ,~ 0. We can use the above to produce a dyadic tree of sets (An,j a~'2n +n=0, j= 1 as in the
supports of the Haar basis and a sequence (fn) as follows. Set f0 -- 1 a0,1 = 1,
fl -- 1Al,l -- 1Al,2, f2 -- 1A2,1 -- 1A2,2
and so on. Then (f,) is 1-equivalent to (hn) and for some subsequence (Pk) of N
O

134 D. Alspach and E. Odell
of (Xn) for some subsequence (Xin) and Ixi* n (Txi~)l ~ e for all n and some fixed e > O.
Then (Txi~) is equivalent to (xi~) and [(Txin)] is complemented in X.
By the lemma (Px(h in,jn)) is equivalent to (h in'jn) and [Px(h in,j~ )] is complemented in
Lp(~2). Also (h i"'j~) is equivalent to (hi~) and this completes the proof.
D
The nonseparable Lp-spaces are built on the spaces gp(F) and Lp{O, 1} ~. If one is
considering only separable subspaces X of such spaces then X ~ L p. But the global
structure of nonseparable L p-spaces can be somewhat different. These spaces need not
have an unconditional (uncountable) basis. Recall dim X = inf{ot: there exists S C X with
S dense in X and card S = o~}.
THEOREM 9 ([37]). Let 1 < p # 2 < oo and let # be a finite measure with dimLp(#) ~>
l'~co. Then L p (#) does not embed into a space with an unconditional basis.
1
3. Sequences in gp and Lp
In this section we look at some special subspaces of ~p and L p which are isomorphic to
gr, for some r. Of course for ~p, the only possible value of r is p itself, but for Lp there
is always r = 2 and for p < 2 an entire interval of possibilities. General sequences and
even unconditional basic sequences in L p are probably too varied to ever be described (see
Section 10, [50], for a particularly peculiar subspace with unconditional basis), however,
near the end of this section we note some results for Orlicz sequences and other spaces
with a symmetric basis. In the previous section we looked at the span of subsequences of
the Haar system. For the unit vector basis of gp, subsequences are not mysterious, and in
this simpler situation we can successfully describe the span of block bases.
Let (ei) be the unit vector basis of ~p. We begin by recalling the a simple fact about
block bases of (ei) which was mentioned in [49, Section 64], [71, Proposition 2.a. 1 ], [44,
Proposition 1.5.3].
PROPOSITION 10. Suppose that (Xn) is a block basis of the unit vector basis of g.p.
[Xn: n E IN] is isometric to g.p with the isometry which is induced by the basis map
Xn ----> ]]Xn lien, n = 1, 2 .....
A normalized basic sequence is said to be perfectly homogeneous if it is equivalent to all
of its normalized block bases. We have thus shown that the unit vector basis of g p has this
property. It is easy to see that the unit vector basis for co is also perfectly homogeneous. It
turns out that this is a very special property.
THEOREM 11 ([101]). Suppose that X is a Banach space with a normalized perfectly
homogeneous basis (Xn), then (Xn) is equivalent to the unit vector basis of g p or co.
We refer the reader to Zippin's paper or the book [71, p. 60], for the proof. There have
been some generalizations of this result in the context of symmetric spaces [9].

Lp spaces 135
Another interesting property of block bases in ~ p mentioned in [49, Section 4] is the fact
that the closed span is always complemented. Indeed, let Xn = ~iEFn aiei be a normalized
block basis of (ei) (here the Fn are finite subsets of I~t and F1 < F2 < ... in the sense
that maxFn < min Fn+l)and define x n = IlXnll -p Y]icFn lai]P-2aie* for each n, where, as
usual, e* denotes the biorthogonal functional to ei. Then (x*) is biorthogonal to (Xn). The
formula
OO
Px -- E x* (x)xn forxeep
n=l
then defines a norm one projection from g p onto the closed span of (Xn), [71, Proposition
2.a. 1 ], [44, Proposition 1.5.3]. This gives
THEOREM 12. If (xn) is a block basis of g.p, then there is a norm one projection from g.p
onto [Xn : n ~ H].
As noted in Section 2, a very similar argument yields that if (Xi) ~ Lp is a nonzero
disjointly supported normalized sequence then (xi) is 1-equivalent to the unit vector basis
of ep and [(xi)] is 1-complemented in Lp.
The analog of Theorem 12 holds for co and, as it turns out, is peculiar to these unconditional
bases.
THEOREM 13 ([69]). Suppose that X is a Banach space with an unconditional bases such
that for every permutation rc of H and every block basis (yj) of (Xjr(n)), [yj: j ~ H] is
complemented, then (Xn) is equivalent to the unit vector basis of g.p or co.
As was noted in [49, Section 4], it is an immediate consequence of Theorem 12 and
standard perturbation arguments that
COROLLARY 14. If Y is an infinite dimensional subspace of g.p, then for every e > O, Y
has a (1 + E)-complemented subspace (1 + e)-isomorphic to g.p.
We turn to some results of Petczyfiski [86] which were important in stimulating interest
in classification problems for complemented subspaces.
THEOREM 15. If X is an infinite dimensional complemented subspace of g.p, then X is
isomorphic to ( p.
A proof using Pe{czyfiski's decomposition method, [86] is given in [49, Section 4].
A more effective but more complex proof is given in [7]. That proof proceeds by directly
showing that a complemented subspace of g p actually decomposes into a sum of finite
n
dimensional subspaces each isomorphic to g p for an appropriate n and then deducing that
the space is isomorphic to g p from the decomposition.
The proof using the decomposition method depends rather heavily on the ability to estimate
the norm of an operator on a g p sum by norms of the restrictions to the summands. In

136 D. Alspach and E. Odell
Section 5 we will see that the fact that this does not work for other types of unconditional
sums leads to a rather delicate approach to understanding the complemented subspaces of
C
Lp. It is not true in general that if X ~ Y and Y ~ X then X is isomorphic to Y [42].
One problem that was open until the 1970's was whether every Banach space contained
g p for some p or co. If we are satisfied with finite dimensional subspaces Dvoretzky's theorem
[33] tells us that for every Banach space X, ~2 is finitely represented in X. Krivine's
theorem [61 ] extends this to basic sequences: Every basic sequence admits a p so that g p
is block finitely represented therein. Going the other way if X is such that for some ~. < cx~,
1 ~< p < ~, every finite dimensional subspace of X X-embeds into ~p m for some m then X
embeds into L p. This is easy via an ultraproduct argument. In any event if we are seeking
some collection of infinite dimensional "atoms" which must live inside of every X, these
must include more than just co and g p, 1 ~< p < cxz, as witnessed by Tsirelson's example
[99,28]. However we do get a positive result for subspaces X of Lp (1 ~< p < c~). In fact
within a certain class of "stable" spaces the conjecture is true in a strong sense. Before we
discuss these results let us define a notion that is a little stronger than finite representability.
DEFINITION 16. Suppose that (Xn) is a sequence in a Banach space X and that Y is a
Banach space with a basis (Yn). We say (Xn) generates the spreading model (Y, (Yn)) if for
every s > 0 and k E N, there is an N such that
(1 +/3) -1 ~aiYi
k=l
i--1
aiXni ~ (1 + s) ai Yi
k--1
for all N < ni < n2 < ... < nk and finite sequences of scalars (ai). (Y, (Yn)) is said to be
a spreading model over X if there is a sequence (xn) in X generating the spreading model
(Y, (y~)) and such that limnl

L p spaces 137
DEFINITION 17. Suppose X is a separable Banach space. We say that X is stable if for
any two bounded sequences (xn) and (yn) in X and free ultrafilters b/and 12 on N,
lim lim Ilxn -+- Ym II - lim lim Ilxn -+- Ym II.
(n)E/~ (m)EV
(m) E'I2 (n) E/~
It is easy to see that all finite dimensional spaces are stable. In the case of Hilbert space
a simple computation with the inner product establishes stability. Indeed, without loss of
generality let (xn) and (Yn) be weakly convergent sequences in a Hilbert space X with limits
x and y, respectively. (We denote the inner product by (., .).) Noting that lim(n)cU Ilxn II
and lim(m)~V [lYm I[ exist and
lim lim IIx~ -+- Ym II 2 -- IIx~ II 2 - Ilym II 2 - 2 lim lim (xn, Ym)
(n)c/g (m)E]2
(n)ELr (m)EV
= 2(x, y)- lim lim Ilxn + Ym II 2 -[IXnll 2 --IlYm 112,
(m)EV (n)EU
(3.1)
we see that X is stable.
Clearly any subspace of a stable space is stable. Maurey and Krivine proved that if X
is a stable Banach space and 1

138 D. Alspach and E. Odell
The convolution of two types r and or, realized by (Xn), bl and (Yn), 12 respectively, is
defined by
r * cr (x) = lim lim IIx + Xn + Ym II,
(n) e/.d (m)cV
where r and o- are realized by (Xn) and L/and (Ym) and V, respectively. If the space X
is stable the convolution is commutative. Furthermore the definition then does not depend
on the particular realizations of r and o- and r 9 cr can be easily shown to itself be a type
on X.
A type r is said to be symmetric if r = -r, i.e., r (x) -- r(-x) for all x. Observe that
if (Xn) is the fundamental sequence of a spreading model which is isometric to g p over X
then the type realized by (Xn) and any ultrafilter/,4 is symmetric.
DEFINITION 19. A symmetric type r is said to be an s if for any non-negative
scalars a, b, ar * br = (a p + bp)I/Pr.
Notice that if (Xn) is a weakly null sequence in s and limn---,~ Ilxn II = P, 1 < p < cxz,
then r(x) = limn~oc Ilxn + xll = (PP + IlxllP) 1/p, for all x e ep, is a symmetric type. Also
(ar,br)(x) = lim lim Ilx + Xn +Xmll
n--+ oc m--+ oo
= (llxll p +aPp p -+-bPpP) 1/p- (a p nt-bp)l/Pr(x). (3.2)
Thus r is an gp-type.
Now suppose that (Xn) is a normalized sequence and L/is an ultrafilter that realize an
p-type r on X. Then for any x e X,
ql II
lim lim ... lim x n t- aixni -- a l'r. a2"c -k . . . . akr(x)
(nl)Eb/(n2) cb/ (nk)cld i--1
- a r(x)- lim x+ a Xn 9
(n)cld
i=1
(3.3)
(3.4)
Thus (Xn) is the generating sequence of a spreading model over X which is isometric to
(~p, (e,,)) over X.
As noted above Krivine, [61], had earlier shown that in any basic sequence the space
p, for some p, or co is block finitely representable. Usually finite representability tells us
little about the infinite dimensional structure of the space. Maurey and Krivine show that
in a stable Banach space Krivine's theorem can be used to obtain much more.
THEOREM 20. Suppose that X is a separable stable Banach space. Then for some 1

L p spaces 139
THEOREM 21. Suppose that (Xn) is a sequence in a separable stable Banach space X
which generates a spreading model which is isometric to (s p, (en)) over X. Then for every
e > 0 there is a subsequence of (Xn) which is (1 + e)-equivalent to the basis of s
The proof of Theorem 21 is quite easy. Given e > 0 we choose suitable ek $ 0 and using
a compactness argument inductively choose a subsequence (x/7 i) of (x/7) satisfying" for all
k and (/~i)'l __V_ [-1, 1],
k-2
~~j2~ + (l~k_l I p -+-I~klP)l/Pxn~_,
j--I
k-2
-- ~ fljXnj + ([flk-I I p + ]flkl p) l/pel
j=l
< 6k
and
k
Z
j=l
fljXnj
k-2
ZfljXnj + flk-lel + flke2
j--I
< 6k.
Using these two inequalities one can iteratively keep combining the last two terms in
y~] ~jxj to obtain the result.
The proof of Theorem 20 is more difficult. One first shows that a symmetric type cr on a
stable space X is an s type for some 1 ~< p < oe iff for all u > 0 there exists 15 ~> 0 with
~r, (otcr) --/3cr. To get the existence of such acr is fairly involved and employs the notion
of conic classes of types.
See the original paper of Maurey and Krivine, [62,44,40] or [23] for proofs. The last
reference proves the existence of s by using an ordinal index approach which is
quite different from any of the others and is somewhat simpler. One uses Krivine's theorem
in a certain manner to obtain a p. Then given e > 0 one considers
He,p(X) -- { (xi)~ ~ X: (xi)~ is (1 + e)-equivalent to the unit vector basis
/7
Of~p}.
This is naturally a closed tree and can be given an ordinal index [ 16]. One shows that the
index is W l and so the tree has an infinite branch.
In general, even for very well-behaved sequences in L p for 1 ~< p < 2, one cannot find
a nice s basis by passing to subsequences. (See [43].) These results have been extended
to weakly stable Banach spaces (defined as in Definition 17 except one also requires (Xn)
and (Yn) to be weakly null) which allows for the case of co [12].
Thus every subspace of a stable Banach space contains a subspace (1 + e)-isomorphic
to s One way to look at these results is to say that every subspace of L p, 1

140 D. Alspach and E. Odell
inquire if there are stronger results than Theorem 20. One such possibility is to replace X
is stable by a condition implied by stability: the set of types on X, r (X), is separable in the
strong topology (that generated by the pseudometrics dM(a, r) = sup{la(x) -- r(x)l : ~ X,
Ilxll ~< M}). Unfortunately the dual of Tsirelson's space [99] satisfies this condition so
the attempt fails. However Haydon and Maurey [45] have shown that if r(X) is strongly
separable then X contains either a reflexive subspace or an isomorph of ~1.
One might also ask whether there are other nice sequence spaces which embed isomorphically
into L p. The approach of types using convolution cannot be used in these cases
because they depend on formulas which must give rise to an ~p basis, [15,72]. The situation
for p > 2 is constrained by the gp - ~2 dichotomy (see Section 4 or [49, Section 4]),
but for 1 ~< p < 2 there is some room to maneuver. Dacunha-Castelle [30], showed that in
order for a symmetric sequence space to embed in L1, the space must be isomorphic to
an average of Orlicz sequence spaces. Additional work was done by Kwapi6n and Schtitt
[64,63], and Raynaud and Schtitt [88]. The proofs of these results are too technical for us
to delve into here, so we will content ourselves with stating a few of them.
THEOREM 22. There is a universal constant K such that if Y is a subspace of L1 with
C-symmetric basis, then there is a probability space (I2, P) and a family of 2-concave
normalized Orlicz functions F~o, o9 ~ I2 such that
(KC)-If~ IlfllF~o dP ~< ]lfll ~< KCfs2 [ifl[Fo~dP
for all f e Y.
Bretagnolle and Dacunha-Castelle [22] showed that each Orlicz space with a 2-concave
Orlicz function embeds into L1. Thus these two results give a pretty clear picture of the
symmetric sequence spaces which can embed in L1 and hence in Lp, for 1 < p < 2, [93].
(See [53] for some additional material on symmetric and unconditional sequences in L1.)
An example of a more general result from [88] is
THEOREM 23. Lr
is isomorphic to a subspace of L1 if and only if there is a p such
that Lr is order isomorphic to a sublattice of L1 (Lp) and X is isomorphic to a subspace
of Lp.
4. Subspaces of L p
As noted in Section 3, if X c s then for all e > 0, X contains Y with d (Y, s < 1 + e and
Y is (1 + e)-complemented in s Not much more can be said about an arbitrary X c s
X need not even have the approximation property, let alone a basis if p ~: 2, [26]. But we
can say something if X has an FDD.
THEOREM 24 ([57]). Let (En) be an FDD for X c_ g~p. Then (En) can be blocked into an
g~ p decomposition (Fn). Thus X ~ (Y~ Fn ) p.

L p spaces 141
We shall sketch the proof which illustrates the blocking techniques developed by Johnson
and Zippin in several papers [57,56] and later further developed [52]. This technique
will have application for subspaces of L p. An extension of this theorem is given in Propo-
sition 31. If A, B and C are positive numbers we write A c B if A ~< CB and B n 1 so that
2-1-1
I1(I - P~2)xll < 62
if X E B(Ei)I2.
Continue in this manner to obtain subsequences (m i) and (n i) of 1~ satisfying the following
mn
for Gn -- (Ei)i=mn_l+l
(a) IIP~.xll < 6k if x E B[ci]i>~+:, k >~ 1, and
(b) I1(I - P~)xll < ek ifx E B(Gz)~: ' , k~>l .
A (possibly finite) sequence (x~) is a skipped block sequence with respect to (Gi) if
there exist integers r l

142 D. Alspach and E. Odell
THEOREM 26. Let 1

Lp spaces 143
The mapping T'[(Xn)] --+ (Y~ Lp(En))p given by Tx -- (x[En) is norm 1. The diagonal
map D(y~ a~x~) -- (a~xn IEn) is bounded with ]]D[] ~< )~ [71, p. 20]. This yields the result.
E3
To obtain the "1 § e" conclusion of (c) we need to pass to a normalized block basis (yi)
of (xi) which is a perturbation of a disjointly supported sequence. For example if p = 1 we
may assume that xi = ui § di where (di) is disjointly supported and bounded away from 0
in norm (hence equivalent to the unit vector basis of g l) and (u i) is uniformly integrable.
This follows from the subsequence splitting lemma.
THEOREM 29. If (fi) CC_ BLi then there exists a subsequence f/ -- ui § di where ]Ui] /~
]di] -- 0 for all i, (u i) is uniformly integrable and (di) is a disjointly supported sequence.
Since a uniformly integrable subset of L l is relatively weakly compact [49, Section 4],
we may assume that (u i) is weakly convergent and hence cannot be equivalent to the unit
vector basis of el. Thus some normalized block basis (yi) of (Xi) is a perturbation of a
block basis of (di) which yields (c).
D
REMARK. While Hilbert subspaces of Lp (2 < p < cx~) are automatically complemented,
for 1 < p < 2 every Hilbert subspace of L p contains a subspace which is complemented
in Lp. Indeed let (xi) be a basis for X equivalent to the unit vector basis of ~2 with biorthog-
1 1
onal functionals (x*). Let Q'Lq --+ X* be the natural quotient map (p + q - 1). Let
Qyi - x* with ([[yi [[) bounded. By Theorem 27(a) it follows that some subsequence of
(y~) must be equivalent to the unit vector basis of ~2 and [(y~)] is complemented in Lq by
the orthogonal projection. It follows that [(x~)] is complemented in L p.
A subspace of L p which is "small" embeds into g p.
THEOREM 30 ([51,46]). Let X c_ L p, 1 < p :fi 2 < cx~.
(a) If2 < p < cxz and ~2 5/-+ X then X ~ gp.
(b) If 1 < p < 2 and there exists K < ~ so that every normalized weakly null sequence
in X has a subsequence K-equivalent to the unit vector basis of g.p then X ~ g.p.
Part (a) has been strengthened [60] to yield that for all ~ > 0, X l_+~ s We shall sketch
a unified approach to proving both parts. Let T~o -- {(nl ..... nk)" k E N and ni E • for
1 ~< i ~< k} be the countably branching tree orderedby (nl ..... nk)

144 D. Alspach and E. Odell
PROOF. By the proof of Theorem 24 (using Lemma 25) it suffices to show that (En) can
be blocked into a skipped gp decomposition, i.e., there exists a blocking (Fn) of (En) and
C < ec so that if (xn) is any normalized skipped block sequence of (Fn), then (Xn) is C-
equivalent to the unit vector basis of g p. C may be taken to be K + 1. By a compactness
argument it suffices to prove the following.
(A) 3nl 'v'xl E S(Ei)~ 3n2 Vx2 G_ S(Ei)~ 3n3 Vx3 E S(Ei)~ ... so that (xi)~ is K-
equivalent to the unit vector basis of gp.
The formal negation of (A) is
(B) Vnl 3Xl E S(Ei),, 1 Vn2 3x2 E S(Ei),~2 Vn3 3x3 E S(Ei),,~... so that (xi) is not K-
equivalent to the unit vector basis of gp.
Since these are infinite sentences we cannot automatically assume that either (A) or (B)
must be true. However by Martin's theorem that Borel games are determined [73] it does
follow that (A) or (B) is true.
Indeed consider a two player game where player (I) chooses n l E N and player (II)
chooses X l E S(Ei)~. Then (I) chooses n2 > max(suppxl), the support being with respect
to the Ei's. (II) chooses x2 E S/E i )n~ and so on. An outcome of this game is a sequence in
~-- {(nl,xl,n2, x2 .... )" Xj E S(Ei)nj and nj+l > max(suppxj) for j EN}.
Player (I) wins if the outcome belongs to
r -- { (n 1, x l, n2, X2 .... ) E ~" (Xi)C~ is K-equivalent to the unit vector
basis of g p }.
The precise meaning of (A) is that player (I) has a winning strategy while (B) means that
(II) has a winning strategy. If ~ is given the relative product topology then r is a closed
subset of ~. Martin's theorem yields then that this game is determined, i.e., either (I) or (II)
has a winning strategy.
But if (B) holds one easily constructs a block tree T E To(X) with respect to (En) so
that no branch is K-equivalent to the unit vector basis of ~ p.
D
By employing some further blocking and perturbation arguments one can extend this
result to
PROPOSITION 32 ([84]). Let 1 < p < oc and let X be a reflexive space. Assume there
exists K < oo so that whenever T E T~o(X) is a weakly null tree then some branch of
T is K-equivalent to the unit vector basis of g.p. Then there exists a sequence of finite
dimensional spaces (Fn) so that X ~ (~. Fn)ep.
PROOF OF THEOREM 30. One first shows that the hypotheses of both (a) and (b) imply
the hypothesis of Proposition 32. This depends upon the fact that X c L p and arguments
like those used to prove Theorem 27.
For example (b) yields that there exists 3 -- 3(K) > 0 so that if (xi) is a normalized
f
weakly null sequence in X then some subsequence can be written as x i -- Yi -Jr- di where

L p spaces 145
Yi /X di -- 0, the di's are disjointly supported and Ildi I1 ~> 6 for all i while (a) yields in
addition that one can take []di ]1 --+ 1. Using this given T ~ T~o(X) one can select a branch
of the tree exhibiting similar behavior and hence is C (K)-equivalent to the unit vector basis
Of~p.
The proof of Proposition 32 yields that if X c Y, where Y is reflexive with an FDD (En)
then one can block (En) into (Fn) so that X ~ (~ Fn)ep. Thus one can block the Haar
basis into an FDD (Fn) so that X embeds into (~ Fn)ep. Since each Fn ~ gp" for some
m n ~ N, the latter space is a subspace of ~p.
ff]
From Theorems 30 and 27 it follows that a subspace X of L p for 2 < p < oc which
is not isomorphic to ~2 nor to a subspace of s must contain ~2 | g p. What additional
conditions on X ensure that X ~ g p ff~ g2 ? The following gives one such result.
THEOREM 33 ([52]). Let 2 < p < oc. Let X be a subspace of Lp which is isomorphic to
a quotient ofa subspace ofg~p • ~2. Then X embeds into g~p 9 g~2.
The method of proof is a more complicated blocking argument which ultimately produces
a blocking (Hn) of the Haar basis so that the natural mapping
P
is an into isomorphism.
The following is open. It concerns the next layer of small subspaces of Lp.
PROBLEM 34. Let X be a subspace of Lp, 2 < p < oc and assume that (~s ~ X.
Does X ~ s 9 s
The situation for subspaces of L p with 1 < p < 2 is of course more complicated (cf.
Theorem 24). But we can say the following.
THEOREM 35 ([93]). Let 1

146 D. Alspach and E. Odell
where d v = ~b dm. Thus since x --+ x is an isometry of L1 with L l(v), the theorem will
follow provided we have
Step 2: Iq (X) < cx~ for some 1 < q ~< 2.
If not one shows that s is finitely representable in X. This yields that Bx is not uniformly
integrable and hence X is not reflexive.
D
5. s 1 < p < cx~, p 7~ 2
Recall that X is a s if and only if there is a constant )~ < ~ such that for each
finite dimensional subspace Z of X there is a finite dimensional subspace Y of X such
/7
that d(Y, s p)

L p spaces 147
realize this space as (y-~n~ 1 Lp (2-n, 2 -n+l) p. Thus by the decomposition method any
complemented subspace of L p which contains L p complemented is isomorphic to Lp. In
particular L p G g.p(F), where F is a finite or countably infinite set, is isomorphic to L p.
It follows that these two spaces are all of the isomorphic types that we get as the range of a
conditional expectation onto a (non-finite) sub-cr-algebra of the Lebesgue sets. A slightly
more general class of projections are those of the form Pf -- gE(fg -1 lsuppg I~) where
g E L p and/3 is a sub-a-algebra. This is the general form of a contractive projection, [10,
100]. These do not give us any new spaces since the range of such a projection is isometric
to L p (#) for some measure #. We have already seen that the sequence of Rademachers or
Gaussians span complemented subspaces isomorphic to g2. It is easy to see then that there
are complemented subspaces of Lp which are isomorphic to ~p @ ~2 and (y~ ~2)p.
A natural question is whether L p can be split into two spaces which are not isomorphic
to L p, i.e., are there complemented subspaces X and Y of L p such that L p -- X 9 Y
but neither X nor Y is isomorphic to L p? As we have seen in the case of L p (and s
(Theorem 7 and Theorem 15) the answer is no. The fact that L p is primary tell us "half"
of the s but little about the far more interesting other "half". Let us now consider
the one complemented subspace of L p which is not a s
Suppose that X is a subspace of Lp which is isomorphic to ~2. By [58] if p > 2, the Lp
and L2 norms are equivalent and the orthogonal projection is bounded. If p < 2 it is no
longer the case that a subspace isomorphic to ~2 is complemented. This was first proved
for certain values of p by using some results from harmonic analysis, [13, p. 52], [94,20].
Later a completely different approach yielded the existence of uncomplemented subspaces
of Lp, for each p, 1 < p < 2, which are isomorphic to g2, [13]. This immediately implies
the existence of subspaces of L p which are isomorphic to e p and not complemented, since
g p is isomorphic to (~ g~)p. For 2 < p < oc, Rosenthal showed that there are uncomplemented
subspaces of g p which are isomorphic to g p, [91, Theorem 6]. Curiously, such a
subspace cannot be the span of a sequence of independent random variables, [31,17]. For
some time it was thought that the situation might be different for p - 1, but in [ 17] Bourgain
shows that there are subspaces of L1 isomorphic to el which are uncomplemented.
If we look at projections there is one very special property of g2 subspaces of L p,
1 < p < 2. While it may not be true that the subspace itself is complemented, as we have
seen, for any subspace X of L p which is isomorphic to g z, there is an infinite dimensional
subspace Y of X which is complemented. If X is a subspace of L p which is isomorphic to
p, then as was shown in Section 3 it must contain a subspace which is complemented (and
necessarily isomorphic to gp). It follows that for a subspace of Lp which is isomorphic to
~p (~ ~2 there is a subspace isomorphic to ~p • ~2 which is complemented in Lp. This
sort of hereditary complementation property does not hold for general complemented subspaces
of L p. Consequently, a special class of operators has been introduced which have
somewhat better properties with respect to g2 subspaces. This concept occurred implicitly
in [91] and then was developed further in several papers, [7,5,6,38,52]. In [58] it was
shown that subspaces of L p, 2 < p < oc, which are isomorphic to/~2 are complemented
by showing that the orthogonal projection acts as a bounded operator on L p. This is an
example of an operator which is simultaneously bounded in both norms on L p. This, as it
turns out, is not really an unusual situation.

148 D. Alspach and E. Odell
THEOREM 37 ([47,53]). Suppose that 1

L p spaces 149
In order to deal with spaces with a norm defined in terms of an L2 norm and an Lp
norm and with similar subspaces of Lp, we will use the terms (p, 2)-bounded, (p, 2)-
isomorphism, etc., to indicate that the map is bounded, is an isomorphism, etc., in both
norms. For example, if T is a map from (~ Xn)(p,2,(w,,)) into Lp, we would say that
T is (p, 2)-bounded with IlTllp,2 ~< K, if there exists a constant K such that IlT(xn)ll2 ~<
2 1/2
K(y~ Ilxnll~w~)
and IIT(x~)llp

150 D. Alspach and E. Odell
The complemented subspaces of (Y]~ ~2)p have also been classified, [83]. It turns out that
only the obvious spaces occur, g2, gp, gp (9 ~2 and (}-~' g2)p. The proof is rather technical
and mainly consists of carefully distinguishing between complemented subspaces which
are isomorphic to complemented subspaces of gp | ~2 and those that contain (y]~ g2)n-
In 1972 Rosenthal proved his inequality and used it to show that there were several other
s
In particular he defined the space X p which had some surprising properties
and led to the construction [21] of uncountably many isomorphically distinct s
Because X p has proved to be a fundamental space we will present some of Rosenthal's
results in detail.
To construct the space X p for p > 2, we let Xn be the one-dimensional space spanned
by 1[0,1] for all n ~ N, and let (wn) be a sequence of numbers in (0, 1] such that for
every e > 0, ~n:wn 0
then (~ Ixn [ 2 Wn) 2 1/2 >~ (infwn)(y]~[xnlP) lip so that the norm is equivalent to the g2 norm.
There is also a combined case where the sequence breaks into two pieces and thus yields
~p (~ ~2.
We have two issues to resolve with X p at this point. How do we see that it is a s
space and how do we see that we really have only one isomorphism class? The solution to
both involves the construction of (p, 2)-bounded projections. (Also note that we can then
define Xq for q < 2 to be the dual of X p where p and q are conjugate indices.) We will
, ( OO
use the representation above for (y]' Xn)(p,2 (w,,)) in Lp 1-1,,=1 [0, 1]), where the X,, are
one-dimensional spaces spanned by a function Xn whose modulus is an indicator function.
For each n E N let x n = ]lxn 1122Xn and notice that Px - y]~n~ (f x n X)Xn is the orthogonal
projection onto the closed span of (Xn) in L2. Thus P is bounded in the Lz-norm. Moreover
by Rosenthal's inequality (1.11)
n=l
~< C max {
n=l
1"
, Ix~,xl IIx. II p 9
n=l
Because P is bounded in the L2-norm and p > 2, the first expression in the max is bounded
by Ilx l] p. To see that the second is also notice that
n=l
x*~x IIx. llp

L p spaces 151
Because Ixn I is an indicator function, Ilxn II 2 - IIx~ II p and thus (Xn Ilxn 1122 Ilxn II p) is a norm
one sequence of mean zero symmetric independent random variables in Lq, where q --
p/(p- 1). Consequently, the sequence is unconditional and has an upper s estimate.
Therefore
n--I
f x, x IIx, 112 2 IIx, II p
oo
- fx~b~x~llx~l12211x~llp
n-- 1
~< IIx IlpC Ib, I q
n--I
for some constant C. This proves that the second expression is also bounded.
The argument just given has been generalized to replace the one-dimensional spaces by
the range of an orthogonal projection which is bounded on Lp. See [38, Theorem 4.8].
THEOREM 39. Let 2 < p < cx~ and let (wn) be a sequence of scalars in (0, 1]. For each
n let Xn be a subspace of mean zero functions in L p such that the orthogonal projection
from Lp onto X,, Pn, is boundedand supllPnll < cx~. Then (~-~ Xn)(p,2,(w,,)) is isomorphic
to a complemented subspace of L p.
The proof of this generalization is very similar to the proof given above. The main point
is to realize that the pairs (x,, x*) can be replaced by orthogonal projections and their
adjoints.
Next we will use a version of the decomposition method to show that there is really
only one isomorphism class for the spaces X(p,Z,(w,,) ) with (w,) thick near 0. To use the
decomposition method we need a suitable infinite sum of spaces. Rosenthal, [91 ], called
this a symmetric sum; in our terminology this is a (p, 2, (1)) sum, i.e., the sequence (wn)
is just the constantly 1 sequence. As we noted earlier a key feature of the s that
is used in the decomposition method is that one can combine a sequence of uniformly
bounded operators on the individual summands to create an operator on the sum. For the
(p, 2, (1))-sum we have a similar property.
LEMMA 40 ([6, Lemma 2.51). If {Xn} and {Yn} are two sequences of subspaces of
Lp(S2, #) for some probability measure #, and there is a constant K and (p, 2)-
continuous operators {Tn} such that I]Tnllp,2

- y*(Eanen)y
152 D. Alspach and E. Odell
affected by redundancy. Thus we can take a thick sequence (Wn) and define Wm,n = Wn
for all n, m in N, to get a space X = X(p,Z,(wm,n)) which is obviously (p, 2)-isomorphic
to the (p, 2, (1)) sum of infinitely many copies of itself. Now we show how to construct
projections which are (p, 2)-bounded.
As Rosenthal discovered one can take a particular combination of any finite set of
weights (Wn)ncF and get a block of the natural basis (en) of X(p,2,(Wn)) and a correspond-
ing functional which is (p, 2)-continuous. If Zn6F It~2p/(p-2) ~ 1, let
( )-'
Y -- E'wn2/(p-2)D~n and y* - E w2P/(P-2) Ell) n2(p-1)/(p-2)e,n.
Now notice that the map
ncF n~F nEF
Pn(Eanen) -
E w2(p-1)/(P-2)an) y
n~F
is (p, 2)-continuous. Indeed,
(~
w2P/(P -2))'(z ll02(p-1)/(P-Z)an ) y
n6F
( z ) '1~ 2~ "~ 2'a~ (Z - 2~-2'
_ ll)2p/(P -2) Wn 113n
ncF n~F n6F
,j2

L p spaces 153
for all k is (p, 2)-equivalent to the natural basis of X(p,2,(vn)) where
v.-( ~,an,2
w.
2)'/2/(
~_~ la.I p
),/p
n~Fk
n6Fk
2/(p-2)
Notice that for the special case an -- ton
)(p-2)/(2p)
Z to2p/(p-2)
n~Fk
the weight vk is
Also if K1 and K2 are positive constants and (Wn) and (W~n) are two sequences in (0, 1]
such that KlWn

154 D. Alspach and E. Odell
and is uniformly p-integrable. Using this Schechtman showed that @kn= 1 X p contains the
k-fold iterated sum
(Z(Z (Z'r')r2
where 2 ~> rl > r2 > ... > rk ~> p, but not a (k + 1)-fold iterated sum. Thus the spaces
1 Xp, k - , 1 2,.. ., are different.
The tensor product has one serious drawback in that the norm of the tensor product of
the projections grows rapidly. (See [6, Chapter 1] for some details.) Thus there was no
obvious way to use this to produce uncountably many different separable/2p-spaces. In
[21,19] a completely different approach using independent sums was used to produce COl
distinct/2p-spaces.
In order to construct/2p-spaces based on independent sums without having difficulties
with constants, Bourgain, Rosenthal and Schechtman introduce an isomorph of L p which
is convenient for the construction. Let T denote the usual dyadic tree and for each node N
of 7- we consider a copy of {0, 1 }, X2N, with the measure of each point one half. Now we
define a certain subspace L of L p on the natural product algebra and probability measure
on I-IN~'T ~(2N, which is generated by those functions which are measurable with respect
to any branch sub-a-algebra. That is, if B is a sequence of pairwise comparable nodes
in the tree, then we consider the sub-o'-algebra generated by sets which depend only on
the coordinates in B and L contains the functions which are measurable with respect to
each such sub-o'-algebras. It is obvious that this space contains complemented subspaces
isomorphic to L p of the Cantor set (with the usual product measure) because the o'-algebra
generated by one branch is isomorphic to the Cantor a-algebra. Less obvious is the fact
that this subspace L is complemented and thus isomorphic to L p.
By considering subspaces of L, Rp, which are generated by functions which depend on
certain subtrees of 7- of order c~, Bourgain, Rosenthal and Schechtman show that there are
uncountably many s
Their argument is actually to some extent an existence argument.
The proof depends on an ordinal Lp-index which they cannot compute precisely
but are able to show grows with the ordinal index of the subtree. In [6] the precise isomorphic
classification of these spaces is given. It is also shown there that these spaces are not
isomorphic to any of those constructed by Schechtman. More details about the ordinal L p
index can be found in [11 ].
For the case p = 1, Bourgain [ 18,19] adapted some of the ideas from [21 ] to produce an
uncountable family of s with the Radon-Nikodym property. The difficulty was
that the analogous spaces for p = 1 are not complemented so Bourgain showed that there
are enveloping/21-spaces which are complemented. Earlier Johnson and Lindenstrauss,
[48], showed by a completely different approach that there are uncountably (actually 2 s~
many separable non-isomorphic s which have the Radon-Nikodym and Schur
properties but do not embed in separable dual spaces.
At this point a complete isomorphic classification of the separable s seems out
of reach. Most of the known relations among the smaller spaces can be found in [38]. The
space X p itself has resisted study, [5]. It is not known whether there are any complemented
subspaces of Xp other than gp, ~2, ~p 9 ~2, and Xp or any other s
except
under additional assumptions such as having an unconditional basis, [52].

L p spaces 155
As we have noted above the general s
are mysterious but the contractively
complemented ones are just Lp(l z) spaces. This raises the question as to what is the size of
the largest constant 9/such that if P is a projection of norm less than y, then the range of
P is isomorphic to L p (#) for some #. Because the norms of projections are linked to the
parameter )~ in the original definition of the s
there is a companion question as
to the largest constant )~ so that if X is a s
then X is isomorphic to Lp(#). One
can also consider the question of the largest constant )~ such that if X is a s
of L p then X is complemented in Lp.
In [32] it was shown that there is a constant )~p such that if X is a finite dimensional
/7
subspace of L p and T is an isomorphism from s with standard basis (e j) onto X with
][TI[II T-1 l[- )~ < )~p then there are disjoint sets and e > 0 such that IlTejla j II > e for
each j. Moreover e converges to 1 as )~ converges to 1. For the case p = 1 this was sufficient
to imply that X is complemented. The main lemma in Dor's proof is the following.
LEMMA 43. Let gl,g2 .... be a sequence of non-negative functions in L1 and let
O

156 D. Alspach and E. Odell
In the case p = 1 it is sufficient to have a uniform bound on the individual errors
[Ifnl[O, 1]\ai II in order to show that the obvious candidate for a projection is bounded. For
p > 1 Schechtman [97] was able to obtain an error estimate of the form
II~ajTejlAj II a()~, e)(~ lajlP) '/p
with a0~, e) --+ 0 as )~ ~ 1 and ~ --+ 1 which gave the complementation result in this
case also. Thus if X is a/2p,Z subspace of L p, 1 < p < c~, with )~ sufficiently close to
1, by passing to a limit of projections on an increasing family of subspaces Xn we get
that X is complemented. (See, for example, [67].) For the case p = 1 passing to a limit is
more problematical because the spaces are not reflexive. Nonetheless it was shown in [8]
that a/~l,l+e subspace of L1 is complemented if ~ is sufficiently small. Recently [41] a
somewhat different approach to these problems for p = 1 has been given which makes use
of a representation theorem of Kalton for operators on Lp, 0 < p

Lp spaces 157
[11]
[121
[]3]
[141
[151
[161
[171
[18]
[19]
[20]
[211
[22]
[231
[241
[251
[26]
[27]
[281
[291
[301
[3]]
[32]
[33]
[341
[351
[36]
[371
[38]
[391
[40]
[41]
[42]
S. Argyros, G. Godefroy and H.R Rosenthal, Descriptive set theory and Banach spaces, Handbook of the
Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001),
to appear.
S. Argyros, S. Negrepontis and Th. Zachariades, Weakly stable Banach spaces, Israel J. Math. 57 (1)
(1987), 68-88.
G. Bennett, L.E. Dot, V. Goodman, W.B. Johnson and C.M. Newman, On uncomplemented subspaces of
Lp, 1 < p < 2, Israel J. Math. 26 (2) (1977), 178-187.
R.P. Boas, Isomorphism between HP and LP, Amer. J. Math. 77 (1957), 655-656.
F. Bohnenblust, An axiomatic characterization of L p-spaces, Duke Math. J. 6 (1940), 627-640.
J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Bel. 32 (1980), 235-249.
J. Bourgain, A counterexample to a complementation problem, Compositio Math. 43 (1) (1981), 133-144.
J. Bourgain, A new class of El-spaces, Israel J. Math. 39 (1-2) (1981), 113-126.
J. Bourgain, New Classes of 12 P -Spaces, Springer-Verlag, Berlin (1981).
J. Bourgain, Bounded orthogonal systems and the s problem, Acta Math. 162 (3-4) (1989), 227-
245.
J. Bourgain, H.P. Rosenthal and G. Schechtman, An ordinal L P-index for Banach spaces, with application
to complemented subspaces of L p, Ann. of Math. (2) 114 (2) (1981), 193-228.
J. Bretagnolle and D. Dacunha-Castelle, Application de l'dtude de certaines formes lindaires al~atoires au
plongement d'espaces de Banach dans des espaces L P, Ann. Sci. l~cole Norm. Sup. (4) 2 (1969), 437-480.
S.Q. Bu, Deux remarques sur les espaces de Banach stables, Compositio Math. 69 (3) (1989), 341-355.
D.L. Burkholder, Martingales and singular integrals in Banach spaces, Handbook of the Geometry of
Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 233-269.
D.L. Burkholder and R.E Gundy, Extrapolation and interpolation of quasi-linear operators on martingales,
Acta Math. 124 (1970), 249-304.
EG. Casazza, Approximation properties, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson
and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 271-316.
P.G. Casazza and B.L. Lin, Projections on Banach spaces with symmetric bases, Studia Math. 52 (1974),
189-193.
P.G. Casazza and T.J. Shura, Tsirelson's Space, Lecture Notes in Math. 1363, Springer-Verlag, Berlin
(1989). With an appendix by J. Baker, O. Slotterbeck and R. Aron.
J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414.
D. Dacunha-Castelle, Variables al~atoires dchangeables et espaces d'Orlicz, S6minaire Maurey-Schwartz
1974-1975: Espaces L p, Applications Radonifiantes et G6om6trie des Espaces de Banach, Exp. Nos. X
et XI, Centre Math., t~cole Polytech., Paris (1975), 21.
L.E. Dor and T. Starbird, Projections of L p onto subspaces spanned by independent random variables,
Compositio Math. 39 (2) (1979), 141-175.
L.E. Dor, On projections in L ! , Ann. of Math. (2) 102 (3)(1975), 463-474.
A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Sympos. Linear Spaces, Jerusalem
(1961), 123-160.
I.S. l~delgtein and R Wojtaszczyk, On projections and unconditional bases in direct sums of Banach spaces,
Studia Math. 56 (3) (1976), 263-276.
M.A. EI-Gebeily, Isometries and e-near isometries of analytic function spaces, Phd.D. thesis, Oklahoma
State University (1984).
R Enflo and T. Starbird, Subspaces of L 1 containing L 1, Studia Math. LXV (1979), 203-225.
R Enflo and H.R Rosenthal, Some results concerning L P (#)-spaces, J. Funct. Anal. 14 (1973), 325-348.
G. Force, Constructions of 12p-spaces, 1 < p -J= 2 < oo, Ed.D. thesis, Oklahoma State University (1995).
(Online) Available: http://arXiv.org/math.FA/9512208 (June 19 2000).
J.L.B. Gamlen and R.J. Gaudet, On subsequences ofthe Haar system in Lp [1, 1] (1 ~< p

158 D. Alspach and E. Odell
[43] S. Guerre-Delabribre and Y. Raynaud, On sequences with no almost symmetric subsequence, Texas Functional
Analysis Seminar 1985-1986 (Austin, TX, 1985-1986), Longhorn Notes, Univ. Texas, Austin, TX
(1986), 83-93.
[44] S. Guerre-Delabri~re, Classical Sequences in Banach Spaces, Monographs and Textbooks in Pure and
Applied Mathematics 166, Marcel Dekker, New York (1992). With a foreword by Haskell P. Rosenthal.
[45] R. Haydon and B. Maurey, On Banach spaces with strongly separable types, J. London Math. Soc. 2
(1986), 484-498.
[46] W.B. Johnson, On quotients of Lp which are quotients oflp, Compositio Math. 34 (1) (1977), 69-89.
[47] W.B. Johnson and L. Jones, Every Lp operator is an L 2 operator, Proc. Amer. Math. Soc. 72 (2) (1978),
309-312.
[48] W.B. Johnson and J. Lindenstrauss, Examples of 121 spaces, Ark. Mat. 18 (1) (1980), 101-106.
[49] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the
Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001),
1-84.
[50] W.B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, Symmetric structures in Banach spaces, Mem.
Amer. Math. Soc. 19 (217) (1979), v+298.
[51] W.B. Johnson and E. Odell, Subspaces of Lp which embed into lp, Compositio Math. 28 (1974), 37-49.
[52] W.B. Johnson and E. Odell, Subspaces and quotients of lp @ 12 and Xp, Acta Math. 147 (1-2) (1981),
117-147.
[53] W.B. Johnson and G. Schechtman, Finite dimensional subspaces of Lp, Handbook of the Geometry of
Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 837-870.
[54] W.B. Johnson, H.E Rosenthal and M. Zippin, On bases, finite dimensional decompositions and weaker
structures in Banach spaces, Israel J. Math. 9 (1971), 488-506.
[55] W.B. Johnson, G. Schechtman and J. Zinn, Best constants in moment inequalities for linear combinations
of independent and exchangeable random variables, Ann. Probab. 13 (1) (1985), 234-253.
[56] W.B. Johnson and M. Zippin, On subspaces of quotients of (y~ Gn)Ip and (y~ Gn)c 0 , Proceedings of the
International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces
(Jerusalem, 1972), Israel J. Math. 13 (1972), 311-316.
[57] W.B. Johnson and M. Zippin, Subspaces and quotient spaces of (y~ Gn)lp and (~ Gn)co, Israel J. Math.
17 (1974), 50--55.
[58] M.I. Kadets and A. Pe~czyfiski, Bases, lacunary sequences and complemented subspaces in the spaces
Lp, Studia Math. 21 (1961/1962), 161-176.
[59] N.J. Kalton, The endomorphisms of Lp (0

L p spaces 159
[71] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, Sequence Spaces, Ergebnisse der Mathematik
und ihrer Grenzgebiete 92, Springer-Verlag, Berlin (1977).
[72] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete
97, Springer-Verlag, Berlin (1979).
[73] D.A. Martin, Borel determinancy, Ann. of Math. 102 (1975), 363-371.
[74] B. Maurey, Sous-espaces compl~ment~s de L p, d'aprks P. Enflo, S6minaire Maurey-Schwartz 1974-
1975: Espaces LP, Applications Radonifiantes et G6om6trie des Espaces de Banach, Exp. No. III, Centre
Math., l~cole Polytech., Paris (1975), 15 (erratum, p. 1).
[75] B. Maurey, Tout sous-espace de L 1 contient un lp (d'aprOs D. Aldous), Seminar on Functional Analysis,
1979-1980 (French), Exp. 1-2, t~cole Polytech., Palaiseau (1980), 13.
[76] B. Maurey, Th~orkmes de factorisation pour les op~rateurs lin~aires gt valeurs dans les espaces L p, Soci6t6
Math6matique de France, Paris (1974). With an English summary, Ast6risque 11.
[77] B. Maurey, Local theory: history and impact, Handbook of the Geometry of Banach Spaces, Vol. 2,
W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), to appear.
[78] P. Mtiller and G. Schechtman, Several results concerning unconditionality in vector valued LP and
H 1 (Un) spaces, Illinois J. Math. 35 (1991), 220-233.
[79] P. Miiller and G. Schechtman, A remarkable rearrangement of the Haar system in Lp, Proc. Amer. Math.
Soc. 25 (1997), 2363-2371.
[80] P.F.X. Miiller, A local version of a result of Gamlen and Gaudet, Israel J. Math. 63 (2) (1988), 212-222.
[81] EJ. Murray, On complementary manifolds and projections in spaces Lp and g.p, Trans. Amer. Math. Soc.
41 (1937), 138-152.
[82] N.J. Nielsen and E Wojtaszczyk, A remark on bases in s with an application to complementably
universal 12~-spaces, Bull. Acad. Polon. Sci. S6r. Sci. Math. Astronom. Phys. 21 (1973), 249-254.
[83] E. Odell, On complemented subspaces of (~-~ 12)lp, Israel J. Math. 23 (3-4) (1976), 353-367.
[84] E. Odell and Th. Schlumprecht, Trees and branches in Banach spaces, Preprint.
[85] E. Odell and Th. Schlumprecht, Distortion and asymptotic structure, Handbook of the Geometry of Banach
Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), to appear.
[86] A. Petczyfiski, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228.
[87] Y. Raynaud, Stabilit~ des espaces d'op~rateurs CE, Seminar on the Geometry of Banach Spaces, Vol. I,
II (Paris, 1983), Publ. Math. Univ. Paris VII, 18, Univ. Paris VII, Paris (1984), 1-12.
[88] Y. Raynaud and C. Schtitt, Some results on symmetric subspaces of L 1, Studia Math. 89 (1) (1988), 27-35.
[89] Y. Raynaud, Sur les sous-espaces de L P (Lq), S6minaire d'Analyse Fonctionelle 1984/1985, Publ. Math.
Univ. Paris VII, 26, Univ. Paris VII, Paris (1986), 49-71.
[90] H.P. Rosenthal, The Banach spaces C(K), Handbook of the Geometry of Banach Spaces, Vol. 2,
W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), to appear.
[91 ] H.P. Rosenthal, On the subspaces of L P (p > 2) spanned by sequences of independent random variables,
Israel J. Math. 8 (1970), 273-303.
[92] H.P. Rosenthal, On the span in L P of sequences of independent random variables. II, Proceedings of the
Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, CA,
1970/1971), Vol. II: Probability Theory, Univ. California Press, Berkeley, CA (1972), 149-167.
[93] H.P. Rosenthal, On subspaces of LP, Ann. of Math. (2) 97 (1973), 344-373.
[94] W. Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203-227.
[95] G. Schechtman, A remark on unconditional basic sequences in Lp (1 < p < ~), Israel J. Math. 19 (1974),
220-224.
[96] G. Schechtman, Examples ofs spaces (1 < p ~ 2 < cx~), Israel J. Math. 22 (1975), 138-147.
[97] G. Schechtman, Almost isometric Lp subspaces of Lp(O, 1), J. London Math. Soc. (2) 20 (3) (1979),
516-528.
[98] S.J. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math.
159 (1-2) (1987), 81-98.
[99] B.S. Tsirelson, It is impossible to imbed l p or c O into an arbitrary Banach space, Funkcional. Anal. i
Prilo2en. 8 (2) (1974), 57-60. (In Russian.)
[100] L. Tzafriri, Remarks on contractive projections in Lp-spaces, Israel J. Math. 7 (1969), 9-15.
[ 101 ] M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. Math. 4 (1966), 265-272.

CHAPTER 4
Convex Geometry and Functional Analysis
Keith Ball
Department of Mathematics, University College London, London, England
E-mail: kmb @ math. ucl. ac. uk
Contents
Introduction ..................................................... 163
1. Convolution inequalities in convex geometry ................................. 163
1.1. The Brascamp-Lieb inequality ...................................... 164
1.2. The reverse isoperimetric inequality ................................... 169
1.3. The reverse Brascamp-Lieb inequality ................................. 171
1.4. Higher dimensional projections ..................................... 173
2. Volumes of sections of convex bodies ..................................... 174
2.1. Vaaler's Theorem and its relatives .................................... 177
2.2. Upper bounds for the volumes of sections of cubes ........................... 181
3. Plank problems ................................................. 182
3.1. Bang's lemma ............................................... 185
3.2. The affine plank theorem ......................................... 185
3.3. Nazarov's solution to the coefficient problem .............................. 189
3.4. Related results and open problems .................................... 192
References ..................................................... 193
HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 1
Edited by William B. Johnson and Joram Lindenstrauss
9 2001 Elsevier Science B.V. All rights reserved
161

Convex geometry and functional analysis 163
Introduction
This article describes three topics that lie at the intersection of functional analysis, harmonic<br /&