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

**Example text**

5. f i n i t e dimension. 7. IT . x2)ΒΆ for x (x;F). i s dense i n C; F be Cm Xl,X2 (X,) Then C: IC : ~n particular , C; m all manifolds, t h a t are ZocaZly of m (X,) I F i s dense i n Cc ( X I x X 2 ; F ) (x,) (XI I):C is dense in IT. E P o i n t w i s e d e s c r i p t i o n o f closures. M I n t h i s s e c t i o n we o b t a i n a p o i n t w i s e d e s c r i p t i o n o f M under some r e s t r i c t i o n s on t h e module T: and t h e space for The F. corresponding d e s c r i p t i o n i s r e l a t e d t o Whitney c o n d i t i o n s .

Otherwise , l e t H = ( X -, V ) n supp(g) . ,yn E H such t h a t H c V U U V Then i t i s enough Yl Yn t o t a k e g, = g hy h 1 Yn Now l e t K be a nonempty compact subset o f X and V1,. ,Vr such t h a t . - ... ... .. be an open c o v e r i n g o f t h a t f o r every compact and that every . c Vi . ,r l = 1 , g, on and d e f i n e B and Wx a , 0 5 gx 5 1 F c K f i n i t e such By compactness t h e r e e x i s t s . u W, L e t Fi = I x e F : W, c Vil XEF Fi i s nonempty. For i e { l , . .

V e T,(X) ,v # 0 , f 0, x E X Since t h e mapping , it follows that , x # y , are I f E Cm(X) : f ( x ) = f ( y ) 1 proper c l o s e d subalgebras; hence c o n d i t i o n (i)h o l d s . Also, g i v e n and ,v . the set o f a l l g 6 Cm(X) such t h a t x E X dg(x)(v)=O i s a p r o p e r c l o s e d subalgebra; hence c o n d i t i o n ( i i ) h o l d s . Conversely, l e t K c X be compact. L e t W c X be open, w i t h Approximation o f smooth f u n c t i o n s Kc W w and compact. 1 . 2 . 2 . Lema.