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

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 307
Conjecture. 5.4. If C is a Banach ball and W(C) < - 1, then
C is a GC-set.
The above conjecture may be made plausible by a supporting
conjecture (5.9 below) and proofs of both conjectures in four classes
of special cases (Section 6). In the general case, I can prove the follow-
ing.
PROPOSITION 5.5. If C is a Banach ball and EV(C) < - $, then
C is a GC-set.
Before proving Proposition 5.5 we introduce another construction
and some other facts. Given a compact Banach ball C in H and an
orthonormal basis {F~}& of H, let F, be the linear span of v1 ,..., ?n ,
and
C, = CnF, (Co = F. = {O}).
Given two sets A and B in H we define their distance as usual,
e(-%B)=s~p$X-YIl’
d(A, B) = e(A, B) + e(B, A).
We shall say the basis {vi> is adapted to C if
4G-, , C) = d(G-, , G)
for 7t = 1,2,... . Since C is compact, a basis adapted to C always
exists. Then the sequence {F,} of subspaces will also be called adapted
to c.
If there is a sequence G,, C G, C +a* of subspaces of H with each
G, n-dimensional and d(C n G, , C) < a, for all n, a, JO, then the
sequence {an} will be called adapted to C (whether or not the G, are).
In order to fmd an upper bound for E-entropies of sets with a given
adapted sequence {a,} we use the following result.
LEMMA 5.6. Let B({ai}&) b e a rectangular n-dimensional block
ofsides2aS,0