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

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

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IIS(~)II' = IlSlt. • Let S and T be a-/~nite measure spaces. Then L ~ ( S ) ®), L~, (T) is a subspace of Loo(S × T). This subspace is usually proper. PROOF. Let X = L I ( S ) and Y = LI(T). Then X* = L ~ ( S ) and Y* = L ~ ( T ) . , Y)* where F R denotes the closure of the set of finite-rank operators. 16, we have Loo(S) ®~ Zoo(T) c [LI(S) ®7 LI(T)]* = L I ( S x T)* = Loo(S x T). The inclusion in (7) is usually a proper one. 34 • CHAPTER 2 PROXIMINALITY Let K be a subset of a Banach space X. We say that K is p r o x i m i n a l in X if to each x in X there corresponds at least one point/co • g such that I I x - k0l[ = i n f k e g [Ix -- kll.

3, there exists a constant c such that each functional ¢ in U ± + V ± has a representation of the form ¢ = ~ + ¢, with ~ E U ±, ¢ E V ±, and [[0[[ + [[¢[[ < c[[¢[]. We apply this to ¢k, getting ¢k = 0 + %b and [[0][ + [[¢I[ -< c. Define ¢k+1 = - - ¢ - 4o. Obviously 11¢k+i 11 _< c + 1 and 4k+1 + 4o = - ¢ 6 V ±. Also tk+x + tk = - ¢ - ¢ o + e + ¢ = e - ¢0 ~ U ±. ' k + l ¢i. i=o ¢ = (¢0 + ¢1) + (¢2 + ¢~) + + (¢k + tk+~) e c ± and ¢ = (¢, + ¢2) + - - . + (¢k-1 + Ck) + (¢k+i + ¢o) e v ±. ~÷kll)} - - 7 - J J ~ l l lim Hx,~ll.

Then for each f E L I ( S , Y ) (list (L PROOF. LI (S, H) ) = Is (list ( f ( s ) , H)&. F o r a n y g E L1 (S, H ) we have, b y t h e definition in C h a p t e r 10, IlS-~I1 = f~ llS(~)- 9(~)llas >_f~ dist (f(8),H)ds. By t a k i n g an i n f i m u m on g we o b t a i n (list ($, fs LI(S, U)) > (list (f(~), H)ds. F o r t h e reverse inequality, let e > 0, a n d let f ' b e a s i m p l e f u n c t i o n in L I ( S , Y ) such t h a t [If - f ' l l < e. ~in=lxiyi where t h e xi are t h e c h a r a c t e r i s t i c f u n c t i o n s of sets Ai in d a n d Yi 6 Y.