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**Example text**

1 . 2 . 2 . Lema. i s t h e mapping gl There e x i s t s . YgN) @= (91 ,. ,.. , gN 6 Cm For every (iii) If b i s the o r i g i n i n RN , then 0 B x W G 0 @(W) + R i t follows that $ : RN + R. - Since 0 f 0 @(w). @ ( W ) C RN n. 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 Cm diffeomorphism f r o m i s c l a s s Cm. i s class n. 2) d @ ( x ) has rank submanifold o f dimension theoremywe have t h a t RN C m - homeomorphism. (ii) @(W) + is a @/W : W + @ : X the following conditions hoZd: (i) Therefore f such t h a t i f G , then Assume t h a t lemma 1 .

Since + , Bc = dim(X) it follows that c B, . Since C: (X) i s dense i n A , it i s enough C! (X) c Bc . L e t f E C F (X) and K = s u p p ( f ) . 7) t h a t f o r e v e r y x E K t h e r e e x i s t s a corn- t o show t h a t Approximation o f smooth f u n c t i o n s V, p a c t neighbourhood condition ( i ) of xl,.. , r erf t ( B n ) l V, . V, Since u K and ei - e Bc eif and by is K ... u ix. t8r i n particular 1 ... ,x f e flV such t h a t we can assume t h a t compact, t h e r e e x i s t By c o n d i t i o n x 35 E , i = 1, ..

Kc X given a compact subset of X and a f i n i t e open covering U1,. ,en which i s a p a r t i t i o n of u n i t y on E A Un ofK, K m A c Cc(X) Let 39 subordinate Proof. t o the given covering. f E C T (X) I t i s enough t o prove t h a t g i v e n such t h a t R[fl = 0 $(O) , then $ f belongs t o t h e c l o s u r e o f t h e a l y e b r a I n f a c t , assuming t h i s , l e t @ E Cm (R) 0 R generated by f. over be such t h a t @ = 0 on , (-m,1/21 . 6 . a r e f u l f i l l e d and lemma Then l e t can assume t h a t .