50

NONLINEAR REPRESENTATION AND SPACES

where Hjq is an increasing continuous function depending only on K. Since p1{T(u)s) —

PAr_i(T(w))= 0 if pN{T(u)) = 0,N 1, it follows from (2.112) that inequality iii) of the

theorem is true for pN(T(u)) = 0. Let pN(T(u)) 0 and let x = \\u\\E /pN(T(u)). Since

pN_1{T(u)) pN{T{u)), inequality (2.112) gives

x 1 +

ANxp~1/2,

x 0, (2.113)

where

AN

=

HN(P1(T(U))).

If x 1 it then follows from the definition of x that inequality

iii) of the theorem is true. If x 1, it then follows from (2.113) that x l + ANx1^2, since

0 p 1/2, which shows that x1/2 AN/2 + ((^iv/2)2 + 1)1/2 2((AN/2)2 + 1)1/2.

S o x A?N + 4 and II^H^ {AN -f 4)pN(T(w)), which proves that inequality iii) of the

theorem is also true for x 1. This proves the theorem.