On the Characters and the Plancherel Formula of Nilpotent Groups ...

JOURNAL OFFUNCTIONAL ANALYSIS 1, 290-330 (1967)
The Sizes of Compact Subsets of Hilbert Space
and Continuity of Gaussian Processes
R. M. DUDLEY*
Department of Mathematics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
Communicated by Irving E. Segal
Received April 18, 1967
1. THE SIZES OF COMPACT SETS
The first two sections of this paper are introductory and correspond
to the two halves of the title.
As is well known, there is no complete analog of Lebesgue or Haar
measure in an infinite-dimensional Hilbert space H, but there is a need
for some measure of the sizes of subsets of H. In this paper we shall
study subsets C of H which are closed, bounded, convex and symme-
tric (- x E C if x E C). Such a set C will be called a Banach ball,
since it is the unit ball of a complete Banach norm on its linear span.
In most cases in this paper C will be compact.
We use three main measures of the size of C. One is as follows.
Let V, = V,(C) be the supremum of (n-dimensional Lebesgue)
volumes of projections P,(C) where P, is any orthogonal projection
with n-dimensional range. Then we define the exponent of volume of C,
J-V), by
1% vn
EV(C) = lim sup ~.
n+cc nlogn
Another numerical measure of the size of C involves the notion
of E-entropy [12]. Let (S, d) b e a metric space. The diameter of a set
T C S is defined as
sup (4% r) : x, y E T).
Given E > 0, one defines N(S, E) as the minimal number of sets of
diameter at most 2~ which cover S. Then the r-entropy of S, H(S, E),
* Fellow of the Alfred P. Sloan Foundation.
290