ABSTRACT
O i , . . . , ftm C i9 s u c h t h a t s u p p (p C U j L i % • L e t x G s u p p 99, t h e n t h e r e e x i s t s a n o p e n n e i g h b o u r h o o d Ux o f x c o n t a i n e d i n flj f o r s o m e j . O b v i o u s l y , w e c h o o s e Ux s o t h a t Ux C ftj. B e c a u s e o f t h e c o m p a c t n e s s o f s u p p (p o n e c a n c h o o s e a finite s e t o f t h e n e i g h b o u r h o o d s U X l , . . . , UXn, X{ G s u p p <¿> s o t h a t s u p p ip C U ¿ = i Í 4 ¿ a n d
C í í ¿ f o r s o m e j . P u t ÜTj : = U l = i UXi. I n v i e w o f L e m m a 2 . 1 . 1 t h e r e e x i s t s a f u n c t i o n ijjj G T>(ftj) s u c h t h a t 0 < ipj{x) < 1 , x G f 2 j a n d ipj(x) = 1 f o r x G ü f j . W e s h a l l n o w d e f i n e n e w f u n c t i o n s a s f o l l o w s :
V?2 ~ ^2(1 - ^ l ) ,
W e a r e a b l e t o s h o w t h a t
k k
j = l j = l O f c o u r s e , ( 2 . 1 . 2 ) i s t r u e f o r k = 1. S u p p o s e
j = i i = i
T h e r e f o r e
3=1 3=1 T h e e x p r e s s i o n o n t h e r i g h t h a n d o f t h i s e q u a l i t y c a n b e w r i t t e n a s f o l l o w s
i i i+l
H(i - ^ ) + W i + i - = i r * - j = i i = i i = i
T h u s , ( 2 . 1 . 2 ) i s s h o w n . N o t e t h a t (p(x) Y[kj=1(l - ipj(x)) = 0 f o r x G O . I n d e e d , i f x G s u p p </?, t h e n 1 — ipj(x) — 0 f o r s o m e j b u t i f x £ s u p p <p t h e n = 0 . T h u s , w e h a v e p r o v e d ( 2 . 1 . 1 ) . •
DEFINITION 2 . 1 . 1 . A c o l l e c t i o n = {fta : a £ A} oí o p e n s e t s fta c W1 s u c h t h a t C U a G A i s c a l l e d a n o p e n c o v e r i n g o f A
THEOREM 2 . 1 . 2 ( S m o o t h p a r t i t i o n o f u n i t y ) . Let A be an arbitrary set in W1 and d be an open covering of A. Then there exists a family ^ of functions I/J in V 'which have the following properties:
( i ) 0 < I¡J(X) < 1 for x G W1, ( i i ) if K (& A, then all except for possibly a large finite number of ip G VP
vanish identically on K, ( i i i ) for each function ip in \& there exists ft G i9 s t¿c/¿ í f t a í s u p p xp C ft, ( i v ) / o r e v e r y x e A, J2^e* 0^*0 = !•
DEFINITION 2 . 1 . 2 . A c o l l e c t i o n \I> o f f u n c t i o n s ^ f r o m D h a v i n g t h e p r o p e r t i e s l i s t e d i n t h e a b o v e t h e o r e m i s c a l l e d a s m o o t h p a r t i t i o n o f u n i t y f o r A s u b o r d i n a t e
t o tf.
