By Jose G. Llavona

This self-contained publication brings jointly the real result of a swiftly becoming region. As a place to begin it provides the vintage result of the idea. The publication covers such effects as: the extension of Wells' theorem and Aron's theorem for the tremendous 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; automated continuity of homomorphisms in algebras of constantly differentiable capabilities, and so forth.

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

**Example text**

F n ) u ~, . . , u ~e T ~~ and and c o n s i d e r . For some constants L * i s o n t o and t h e r e f o r e i t i s an 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 theorem we have t h a t isomorphism. ,... ,k we have L(hzU2) = i = 1,2 Otherwise we remark be such t h a t no i s t h e canonical b a s i s i n Rn L(hlul) i s a local such t h a t I n t h i s way we o b t a i n ... ,en 8. -1 . ) = 0 , 1 5 i < j < n L e t f, = ( f l J t h e l i n e a r mapping L :. , f n 6 G such t h a t d f i ( x ) ( u i ) and XI I n particular, there exist = 0 dfz(x)(up) # 0 such t h a t f x = fl.

2 . 2 i s t r u e . is a 27 Cm. If f W to @(W). Cm(X), t h e mapping 6 From W h i t n e y ' s e x t e n s i o n theorem @-I : @ ( K ) -+ R extends t o a Cm - f u n c t i o n Therefore, +(D) = d @(W) we can assume t h a t From W e i e r s t r a s s ' 0. theorem 1 . 1 . 2 i t f o l l o w s t h a t t h e r e e x i s t s a sequence o f p o l y n o m i a l s pr(xl,. ,xN) k apr (*) Since such t h a t : r+m pr(0) + s t a n t term. ,gN) , f r o m a l l k e Nny I k l pr 5 i s without con (*) i t follows that: u n i f o r m l y on K, f o r a l l k e Nn , Ik r +m Since each fr e R[Gl t h e p r o o f o f theorem 1 .

A constant all Let (Vi,@i) = f o $; Hence f o r e v e r y ciYk supp(q o f) i t follows that E Ac(X) 1 . 2), k E Nn such t h a t such t h a t 1 3 k ($ Ohi)l 5 c S i n c e t h e s e t o f a l l such c;,~> 0 such t h a t letting k and put i s f i n i t e , we have and a l l i t s p a r t i a l d e r i v a t i v e s up t o t h e o r d e r Si(Vi). 6) f E R[fl f(K). . n = dim @i(Vi) that q K, and on E E are contained i n We by W e i e r s t r a s s ' theorem t h e r e e x i s t s a polynomial q, w i t h o u t c o n s t a n t term such t h a t I @ q l 5 1 Hence - I $ 0 f - q o f l 2 1 on X.