On the Characters and the Plancherel Formula of Nilpotent Groups ...
On the Characters and the Plancherel Formula of Nilpotent Groups ...
On the Characters and the Plancherel Formula of Nilpotent Groups ...
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SIZES OF COMPACT SUBSETS OF HILBERT SPACE 295<br />
for all s in T, <strong>and</strong> for any t in T such that d(s, t) < 1/2no+a, we have<br />
I L(s) (w) -L(t) (w> I < 8.<br />
Letting 6 4 0, we see that L is uniformly continuous on T with<br />
probability 1. Q.E.D.<br />
COROLLARY 3.2. If S is a subset <strong>of</strong> a Hilbert space <strong>and</strong> r(S) < 2,<br />
<strong>the</strong>n S is a GC-set.<br />
There are numerous examples <strong>of</strong> sets S with 7(s) = 2 which are<br />
nei<strong>the</strong>r GC- nor GB-sets; see, e.g., Section 6 below. Moreover,<br />
Theorem 7.1 below <strong>and</strong> its partial converse, due to Fernique [7],<br />
indicate that, even when specialized to stochastic processes on <strong>the</strong> real<br />
line, Theorem 3.1 is essentially <strong>the</strong> best possible result <strong>of</strong> its kind.<br />
However, we prove in Proposition 6.10(a) below that no sufficient<br />
condition for <strong>the</strong> GC-property <strong>of</strong> a Banach ball in terms <strong>of</strong> H(S, E)<br />
is necessary, i.e., <strong>the</strong> GC-property is not a “monotone function” <strong>of</strong><br />
<strong>the</strong> function E + H(S, l ) as E J 0. Yet I make<br />
Conjecture 3.3. If S is a GB-set (<strong>and</strong> hence if S is a compact<br />
GC-set), <strong>the</strong>n r(S) < 2.<br />
In Sections 6 <strong>and</strong> 7 below, Conjecture 3.3 is proved in a number<br />
<strong>of</strong> special cases. In <strong>the</strong> general case, I shall prove at present only <strong>the</strong><br />
following:<br />
PROPOSITION 3.4. If S is a GB-set <strong>the</strong>n S is totally bounded<br />
(i.e., its closure is compact).<br />
Pro<strong>of</strong>. If S is a GB-set, it is certainly bounded. Suppose it is not<br />
totally bounded. Then for some E > 0 <strong>the</strong>re is an infinite sequence<br />
{ fj}& in S such that <strong>the</strong> distance <strong>of</strong> fi+l from <strong>the</strong> linear span Fi <strong>of</strong><br />
fi ,..., fi is at least E for all j. Let<br />
where ]I g, I/ >, E <strong>and</strong> g, 1. F, . Given M > 0, let<br />
Then<br />
A,={w:max(lL(f,)I:l~j~n} WI<br />
= Pr W5J > M) Pr &J/2.