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By Jose G. Llavona

This self-contained publication brings jointly the $64000 result of a quickly growing to be sector. As a kick off point it provides the vintage result of the idea. The booklet covers such effects as: the extension of Wells' theorem and Aron's theorem for the effective topology of order m; extension of Bernstein's and Weierstrass' theorems for limitless dimensional Banach areas; extension of Nachbin's and Whitney's theorem for countless dimensional Banach areas; computerized continuity of homomorphisms in algebras of always differentiable features, and so forth.

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In t h i s . Thus K , ( i ) hoZds case y i s not necessariZy resuZt (11 valued Bore1 measures. 26 ; 3 . 2 7 ) . 23 Chapter 1 APPROXIMATION OF SMOOTH FUNCTIONS ON r1ANIFOLDS T h i s c h a p t e r begins w i t h a s e c t i o n d e d i c a t e d t o W e i e r s t r a s s ' theorem on t h e a p p r o x i m a t i o n o f d i f f e r e n t i a b l e f u n c t i o n s by Dolynomials. 2, 1 . 4 a r e c e n t e r e d around t h e problem o f d e s c r i b i n g denses subalgebras i n t o p o l o g i c a l a l g e b r a s o f d i f f e r e n t i a b l e f u n c t i o n s .

Let T be a continuous mapping from a realcompact space (4) X i n t o rz space of Y is realcompact. 9 , 8). Holomorphic f u n c t i o n s . U be an open subset o f a complex Banach space E . Let tion Then t h e t o t a l preimage of each realcompact subset Y. f : U C + A func- i s s a i d t o be G-hoiomorphic (Gateaux-holomorphic), if t h e f u n c t i o n d e f i n e d by where D = { A e IC : x t Ay E U). A funcx,y a E i s s a i d t o be holomorphic i f i t i s G-holomorphic and i s a n a l y t i c f o r every tion f : U + t continuous.

2 . 2 . Lema. i s t h e mapping gl There e x i s t s . YgN) @= (91 ,. ,.. , gN 6 Cm For every (iii) If b i s the o r i g i n i n RN , then 0 B x W G 0 @(W) + R i t follows that $ : RN + R. - Since 0 f 0 @(w). @ ( W ) C RN n. By a p p l y i n g t h e i n v e r s e f u n c t i o n Cm diffeomorphism f r o m i s c l a s s Cm. i s class n. 2) d @ ( x ) has rank submanifold o f dimension theoremywe have t h a t RN C m - homeomorphism. (ii) @(W) + is a @/W : W + @ : X the following conditions hoZd: (i) Therefore f such t h a t i f G , then Assume t h a t lemma 1 .

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