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Extra resources for Approximation Theory and Functional Analysis 1977: International Symposium Proceedings
T h i s can be s e e n a l s o d i r e c t l y by means of t h e r e p r e s e n t i n g measures. T h i s d e s c r i p t i o n of ber [a,b] A Xu makes thenun- of i t s dimension e v i d e n t . 111. RELATIVE KOROVKIN APPROXIMATION W e r e t u r n now t o t h e s i t u a t i o n s t u d i e d i n C h a p t e r I. Hence i s a f u n c t i o n s p a c e on a compact m e t r i z a b l e s p a c e X. of h e l a t i v e Korovkin approximation i f t h e r o l e o f X W e s h a l l speak C ( X ) in absolute Korovkin approximation i s t a k e n o v e r by a c l o n e d dunctian n p a c e d: c o n t a i n i n g JC as l i n e a r subspace: J € c d: Consequently, a sequence (JC, f ) - = d: c C(X).
26]). p. I26 E BE F of Grothendieck E E F. W e c a n now f o d a t e SchwatLtz'b , Proposition 11, c f . 9, and [ 8 ] ) : THEOREM (L. c. bpace F T h e quahi-complete L . c . i n denbe i n E 0 F equivalently, (at, and F ahe complete l . c . get: bpaCeb E EF doh bpace E ha4 t h e a . p . d o h each ( q u a s i - ) c o m p l e t e each Banach Apace F ) . S o id buch t h a t E o h F han t h e a . p . , V E E F = E BE F, t h e c o m p l e t i o n 06 t h e E - . t e M b O t phoduct ( w h i c h we w i l l UehO caLC, doh E we E QE F b h a h t , c o m p l e t e E-tenboh p h o d u c t ) .
1) u s i n g , among o t h e r ( o b v i o u s ) t h i n g s , t h a t f i n i t e E - p r o d u c t s are U h A o C i a t i W e a n d t h a t E - p r o d u c t s of carrplete spaces w i t h a . p . p. 9 . COROLLARY: with x1 nheaveb ... x OA xn x - c xll.. hen p e c t i v e l y . ,Xn X1,.. 4li0h,46) bpaCeA a klR-npace, , 'CXn d o h each F1, bet .. ,Kn K1,. p. y i e l d s now: , . , Fn compact hubbe& 06 06 neighbouhhoodn 0 6 Ki Ui E Uli ( i=1, a t mod2 one i) o h l e t a l e b u t one have t h e a .
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