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

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 297
from Rk into itself with norm 11 A (1 < 1 and let C be a convex symmetric
set in Rk. Then
G(A(C)l) > G(C).
Proof. This follows directly from [9], Theorem 5, stated in dif-
ferent language. For A symmetric and invertible it is Lemma 5.2 of
[9], and arguments to reduce to this case are given in the proof of
Theorem 5. Q.E.D.
Now as usual, let H be a separable, infinite-dimensional Hilbert
space. Every GB-set in H is included in some Banach ball which is
still a GB-set.
For any subset C of H, let s(C) be its linear span. Then if C is
convex and symmetric, it is the unit ball of a norm 11 * ]Ic on s(C). If
C is a Banach ball, then (s(C), II l Ilo) is a Banach space and its natural
injection into H is continuous.
Let H* be the dual space of H. (For clarity, we do not identify
the two.) For each 93 in H*, let
II 9J II;: = SUP {I d$) I : $J E 0
Then if C is a Banach ball in H, 11 * 11; is the dual norm to II * lIc
(composed with the natural map of H* into the dual space
MC)‘> II * IILL) of (4Ch II l IICN.
L is an assignment of random variables to elements of H, or equivalently
to continuous linear functionals on H*. The assignment can
be extended to some nonlinear functionals in various ways. For
example, if q~ is a Bore1 measurable function on R” and fi ,..., f, E H,
then ol(fi ,-.., f,) defines by composition a function on H*. (Such a
function is called “tame.“) The assignment
L(dfl 9.*-Y fn>) = dL(fl)Y.,L(fnN
is well-defined, as is well known [9], [18]. Thus, e.g., we let
L(lf I> = IL(f) I ,f EH-
Now in general, an assignment such as
L (SUP &) = SUP b%Ad
will not be well-defined, but if g, = If, I , f, E H = (H*)*, then
sup g, = II * 11; where C is the closed symmetric convex hull of the
f, . Also
SUP L(&) = SUD (1 L( fn) I) = acj,