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Example text
When for short, A(K,E) , H(K,E) i n s t e a d of 0 on t h e prod- AF(K,E) , =o, is j u s t we w i l l the write, HF(K,E), respectively. F = O , some o f the r e s u l t s i n s e c t i o n s 1 and 2 are appar- e n t l y p a r t o f t h e “ f o l k l o r e “ of t h e subject, b u t u s u a l l y n o t easily For e have a l r e a d y p o i n t e d o u t i n t h e i n accessible i n the l i t e r a t u r e : W t r o d u c t i o n t h a t this p a p e r i s b a s e d on a g e n e r a l i z a t i o n of t h e “ o l d ” article [ 2 1 .
C resp. [12 ], instead of Arhangel'skir's [ l o ] , 1 . 1 0 ) t h a t t h e "product VECTOR-VALUED APPROXIMATION ON COMPACT SETS F1 E i s u n i q u e L y d e t e h m i n e d by t h e f o l l o w i n g r e q u i r e m e n t s : F2 For a l l open subsets U. Ui a n d , f o r a l l open r 41 F1 F2 u1 U2'V1 C on 3 Vi v2 ( i= 1 , 2 ) , Xi Xi = rF 1 ulvl (F1~F2)(Ll1xU2) F 1 (U1) € F 2 ( U 2 ) , = ( i=1,2), ~r F2 u2 v2 F rUVd e n o t e s t h e c a n o n i c a l r e s t r i c t i o n mapping where F and where t h e € - p r o d u c t with r e s p e c t t o t h e sheaf l i n e a r mappings i s d e f i n e d i n , s a y , [ 7 ] .
E s p e c i a l l y , the sheaf X of harmonic f u n c t i o n s on IRn satisfies a l l a s s u m p t i o n s o f 1. (ii)above, and a l s o t h e "harmonic s h e a v e s " o f a b s t r a c t p o t e n t i a l t h e o r y are s h e a v e s of F-morphic f u n c t i o n s . All t h e s h e a v e s of example 1. a r e (FN)-sheaves. 2. e. ( i i i )H F ( K , E ) {f E := e'of I I f(EFE(Ei)r f K E the closure i n C(K,E); (depending on 0 F ( K ) f o r e a c h e'E E ' } , and C ( K , E ) of t h e r e e x i s t s an open neighbourhood f ) and a f u n c t i o n g c o n t i n u o u s and e ' o g E F(U) for any e'E E U of K E F ( U ) [ i .