By Jose G. Llavona

This self-contained e-book brings jointly the real result of a speedily growing to be quarter. As a place to begin it offers the vintage result of the speculation. The ebook 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 limitless dimensional Banach areas; extension of Nachbin's and Whitney's theorem for limitless dimensional Banach areas; automated continuity of homomorphisms in algebras of continually differentiable capabilities, and so forth.

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5) i s dense i n , A = F'o M Conversely, l e t From (X)) # 0 (C: x when , v e X E , v # 0 Tx(X) i t follows that (No). satisfies condition A. , dh(x)(v) B R ,v 1 eF' A l s o t h e mappings - Hausdorff, t h e a l g e b r a over , (X) f E C: o C, = (X) a r e continuous. + A N = A I F c M. 2 and theorem F Since i s s t r o n g l y s e p a r a t i n g and N is i s a module i t follows that N, hence M, C,(X;F). rn 2 1 Now l e t . 3 m 'I -dense i n Cm($(V) ; F) f o r a l l (V,$) e Ac(X) . From ( 1 . 5 .

3 ) and ( 1 . 4 . 4 ) we have we have (C: (X) - aZgebra. 5. CorolZary. (c: (x) (b) Let A E ,T y ) is a Nachbin m-algebra Toprn [X) a ,c . Then CF(X) is dense i n A . CoroZZary. Let X and Y be Cm manifoZds of f i n i t e dimension. 6. If A (a) 8 Toprn ( X x Y), a ,c then Cy(X) @ Cy(Y) i s dense i n A. 7. - supporting family, providing admissible topological algebra w i l l algebra. Let A be a topological space of functions defined Definition. ) of nonempty closed J jEJ X i s called a supporting f a m i l y f o r A i f given B c A closed subsets of we have f E B f E A a d m 37 when f I Xj E ,j BIXj E .

7 R [go,. ,g,,)c Then A i s a Nachbin rn Proof. L e t A . ygn A. 2). (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 )..