By William Allan Light, Elliott Ward Cheney (auth.)

ISBN-10: 3540160574

ISBN-13: 9783540160571

ISBN-10: 3540397418

ISBN-13: 9783540397410

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7 T H E O R E M . Let S and T be closed and bounded intervals of the real line. Then there exists a one-dimensionM subspace H in C(T) such that C(S) ® H is not proximinal in C ( S x T). PROOF. T h e c o n s t r u c t i o n and p r o o f are simpler ff we use S = [ - 1 / 2 , 1/2] and T = [--1, 1]. -"1) (s,t) e S x T. 37 One can show t h a t the best a p p r o x i m a t i o n to fe is h if s > 0 and - h if s < 0. F r o m this, dist {fs, H} = 1 - Ish where H denotes the subspace generated by h. It then follows t h a t dist ( F ~ , H ) = 1 for all s, and dist ( F , C ( S ) ® H ) = 1.

0. A m a p p i n g F defined on a subset of a normed space is said to be n o n e x p a n s i v e if [ [ r x - r y l l < [ I x - y[[. It is said to be o d d if F ( - x ) = -Fx. The following theorem is due to Baillon, Bruck, and Reich [19]. 6 T H E O R E M . Let C be a s y m m e t r i c , dosed, convex set in a u n i f o r m l y c o n v e x Banach space. Let T : C --~ C be a m a p which is o d d and nonexpansive. If ~ is a point in C for which lim,~(T'~ - T n + l ~ ) = O, then the sequence {T"~} converges to a fixed point ofT.

It suffices to prove this for w E ~ ( P ) ®~ Y and then use a symmetry argument. ~(P) and v E Y . Calculating the required inner product, we have (x®y-Px®y-x®Qy+Px®Qy, g®v> = (y, v) - + (Px, g> (Qy, v) = < x - P x , g> ( y , v > - ( x - P x , g) (Qy, v> =0 because x -- P x ± g. I In previous theorems (viz. ~ nl (T) = L1 (S x T) (iii) L2(S) ®f~L2(T) = L2 (S x T). In the remainder of this chapter, the analogous result for Lp(S x T), 1 < p < oo, will be developed.

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Approximation Theory in Tensor Product Spaces by William Allan Light, Elliott Ward Cheney (auth.)

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