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Extra resources for Approximation Theory and Functional Analysis 1977: International Symposium Proceedings
So l e t f E HF(K,E). function g s&watz's must b e v e r i f i e d f o r t h e f i r s t a s - p be a c o n t i n u o u s seminorm E FE(U) such t h a t = E on compact subsets o f a, HF(K,E) on E, By d e f i n i t i o n , t h e r e e x i s t s a n open s e t definition, FE(U) U E in E F(U) m~ p ( f ( x ) - g(x)) < E > 0 u 3 K and $. B u t , a g a i n and a by ( w i t h t h e t o p o l o g y o f uniform convergence U). p. of F ( U ) or o f E and o n e theorem from t h e i n t r o d u c t i o n imply t h a t direction of E 0 F ( U ) i s dense 44 EIERSTEDT 5.
4) Undeh t h e ahnumptionn o d ( 2 1 , (5) Undek t h e annumptionb 06 ( 3 1 , we h a v e A(K,E) = H(K , E ), too. 11. ( 3 ) i s r e l a t e d t o a r e s u l t of Weinstock [30 I , p . 812, where, i n s t e a d of the assumption of a smooth boundary i n 11. (1)( i i ) ,he needs o n l y t h e s o - c a l l e d " n e g m e n t p h a p e h t y " o f K. ) A t t h i s p o i n t , a few remarks on p a p e r [ 271 a r e a l s o i n o r d e r ( i n c o n n e c t i o n w i t h OLX Sibony's preceding results): P r o p o s i t i o n 1 o f [ 2 7 1 i s , i n some s e n s e , e a s y , i f n o t t r i v i a l , a s o u r theorem 4 .
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 ) .
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