48

NONLINEAR REPRESENTATION AND SPACES

where \Z±\ + \Z2\ = \Z\,\Zj\ 0 , n 2 .

We now prove the second statement of the lemma by induction in \Y\, Y = XZ,

using formula (2.96). If \Y\ = 0 then the statement is true since TJ1 = 0 for n 2. Let

\Y| = L + 1. Then, supposing inequality ii) of the lemma true for all \Y\ L, we get

\\T1XTE(®7=IUJ)\\EN \\TS(^=1Uj)\\Efi+i (2.108)

C|Z|,AT,n ( X , H^ll^+|z|

H^2

H^i I I 11^ WE)

i 1=3

(EKIk+1+wIKII^IllK^)3Z=

p-1/2

It now follows from equality (2.96) and inequalities (2.107) and (2.108) that inequality ii)

of the lemma is true. This proves the lemma.

According to Lemma 2.19 Ty, Y £ II', has continuous extensions to spaces larger than

EQQ. These extensions are denoted by the same symbol Ty.

Remark 2.20. Let 7 € II' and N 0. Then Ty is a continuous linear map from

g EN+\Y\ to EN.

Lemma 2.19 immediately gives estimates for the polynomial Ty.

Corollary 2.21. Ty,Y G II' is a continuous polynomial from EN+\Y\ t° EN satisfying:

i) l|7V(«)||Ejv C

w

,

m

(H

E

)|H|

£ j v + m

,

") l|Ty(«)||Bw CN,m(\\u\\E)\\u\\E\\u\\EN+tyi,

iii) \\TY(U)\\EN C ^ i y i d l u l U J I I t t l l ^ d l u l l ^ ^ ^ ^ )

3

/

2

- " ! ^ ! ! ^ ^ ,

where CN,\Y\ ^S an increasing continuous function and Ty —Ty — TY.

Let Y i— • aY be a linear function from U(p) to a Banach space B. We introduce

(cf. (2.38) of [28])

PN

(*)=•( £

II^HB)

, N0. (2-109)

Yen'

\Y\N

When B = E we write pN instead of p^. According to definition (1.6a) of || • H^ we have

PN(T1(U)) = \\U\\EN, N0. (2.110)

We can now prove that the linear representation T 1 of U(p) is bounded by the nonlinear

representation T, and vice-versa, on a ^-neighbourhood of zero in E^.