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.
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