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

296 DUDLEY
Now for some S > 0, we have, for all n,
Pr (L&) > M) > 28; so Pr (A,) < (1 - S)+l
by induction. This contradicts the fact that S is a GB-set and com-
pletes the proof.
The method of proof just used will yield a stronger result. Using
also (5.2) and Lemma 5.6 (cf. also Proposition 6.9), it can be shown
that, if S is a GB-set, then for any 6 > 0
N(S, c) < exp (exp ( l/~z+s))
for E sufficiently small. Since the examples in Section 6 indicate that
this inequality has an unnecessary extra exponentiation, no further
details will be given.
4. PSEUDO-NORMS
Let V be a real linear space and let W be a linear space of linear
functionals on V. Then for any set C C V, the polar Cl is defined by
When C is symmetric,
Cl = {w E W : W(X) < 1 for all x in C}.
C1 = {w E W : 1 w(x) / < 1 for all x in C}.
If A is a linear transformation of V into itself and W is closed under
the adjoint A* (i.e., composition with A), then for any CC V,
A(C)1 = (A*)-1 (Cl).
(Here (A*)-l is a set mapping and A* need not be invertible.) In
particular V may be a Hilbert space and IV its dual space, possibly
identified with V.
On K-dimensional Euclidean space Rk, let h or A, be Lebesgue
measure and let G be the standard Gaussian probability measure;
dG = (27r)ekj2 exp (- r2/2) dX,
where r is the distance from the origin.
PROPOSITION 4.0 (Gross [9]). Let A be a linear transformation