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By Jose G. Llavona

This self-contained booklet brings jointly the real result of a speedily starting to be region. As a place to begin it offers the vintage result of the idea. The booklet covers such effects as: the extension of Wells' theorem and Aron's theorem for the superb 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 limitless dimensional Banach areas; automated continuity of homomorphisms in algebras of consistently differentiable capabilities, and so forth.

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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.

3) R" , t h e n (Ti) Ac = (Vi,~i)iEI If i s supporting f o r A. I t i s c a l l e d the t r i v i a l supporting family f o r for X Vx there e x i s t s Vxn X j * Let of x x1 ,. such t h a t . , 6, E E x E , there exists K = supp(f)n X flVx E K such t h a t C: j (Bn))Vx K c Wx ' . By compactness * - . J be a p a r t i t i o n o f u n i t y on K where 38 Chapter 1 subordinated t o t h e c o v e r i n g h = Blh t ... t implies t h a t orh (B Bih E and (En) c :C ).. ,Wx Wxl B r . If h = f C F ( E n ) ) \ Xj.

10 , ,0) B c C,(X) Since CiY1(B) c B +(O,.. e Tx(X) , v # 0 , we can t a k e 01 and BP = {g 6 A : g(x) = 0 1 . and = v , then . 4. Nachbin m-algebras. The Nachbin m-algebra concept i s i n t r o d u c e d i n t h i s s e c t i o n , and Nachbin's theorem i s extended t o t h i s new c o n t e x t Nachbin t y p e theorems f o r t h e a l g e b r a s (CF (X), Sm(Rn) . More , (Cy T;) specifically, (X) , T! ) and a r e obtained. 1. Definition. C" m a n i f o l d o f dimension n. denotes a r e a l G c Cm(X) s a t i s f i e s conditions A subset (N) i f ( i ) G i s strongly separating.

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