By Jose G. Llavona

This self-contained e-book brings jointly the $64000 result of a quickly becoming sector. As a kick off point it offers the vintage result of the idea. The e-book covers such effects as: the extension of Wells' theorem and Aron's theorem for the nice topology of order m; extension of Bernstein's and Weierstrass' theorems for endless dimensional Banach areas; extension of Nachbin's and Whitney's theorem for endless dimensional Banach areas; computerized continuity of homomorphisms in algebras of constantly differentiable capabilities, and so on.

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**Extra resources for Approximation of Continuously Differentiable Functions**

**Sample text**

F 6 Fix 2 1 and F has the approximation property. f 6 C F (X;F) Then f o r a l l Proof. Assume t h a t m F'o f B F , f E C F (X;F) and l e t a f i n i t e r' c rm be g i v e n . We can 45 Approximation o f smooth f u n c t i o n s m assume t h a t i s f i n i t e and . a E cs ( F ) f i n i t e and Let r' I ' x {a} , where = {ml x , E Ac(X) (Viy$i) i E I' I 'c I i s , be t h e cor- responding s e t o f c h a r t s . (X) , n k E N, , E A, The f o l l o w i n g o b s e r v a t i o n can be made: g i v e n (V,$) then ak(f @ - l ) ( $ ( V ) ) i s r e l a t i v e l y compact i n 0 F, f o r a l l n = dim $ ( V ) .

6. Dense polynomial algebras. I n t h i s s e c t i o n we o b t a i n dense polynomial a l g e b r a s i n (C: ,in a s i m p l e way, a d e s c r i p t i o n of (X;F) i T; ) r e l a t e d t o Stone and Nachbin conditions. Let X be a l o c a l l y compact H a u s d o r f f space and l o c a l l y convex H a u s d o r f f space. When m 2 1 ,X Hausdorff m a n i f o l d l o c a l l y o f f i n i t e dimension. s i d e r e d on every x 6 C T (X;F) is 7 7 F denotes a r e a l a real Cm The t o p o l o g y t o be c o n .

However t h e q u e s t i o n remains whether or not. 3 Proposition. Then A E TOP! YC (X). Let A B Top! (X) be such t h a t Topm ( X ) = Topr(X) a ,c rn T ~ ( C : (X) <_ -ri . rAICC(X) = Proof. L e t I $ ~ e C;,l(R) bourhood o f 0 I$ E C F (Rn) m in C: in X q and e CF (K) f and go, ... ,gn . Then WK pjyl(f) < m E Cc ( X I E and u K = supp(go) n C: (K) W Jo c I ( K ) . Hence t h e r e e x i s t s such t h a t Let be g i v e n T~ i s a compact subset o f 0 31 W T m i. a neigh- ... u supp(gn) i s a neighbourhood o f f i n i t e , 1 E R1, and imply t h a t j e Jo > 0 E f 8 WK be a polynomial on Rn+1 w i t h o u t c o n s t a n t term and d e f i n e .