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^i I I 11^ WE)
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
£ 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 ^ ^ ^ ^ )
- " ! ^ ! ! ^ ^ ,
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 )
, N0. (2-109)
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^.