By G. Coeuré (Eds.)

Coeure G. Analytic capabilities and manifolds in endless dimensional areas (NHMS, NH, 1974)(ISBN 0444106219)

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Y , [XI < 1 i s an a l g e b r a , t h e same i n e q u a l i t y f o r powers o f If(x+a)[ 6 ( 1 - X I fa' i s continuous + i n c h a p t e r I , and ( b ) i s j u s t p r o v e d . W , all f k , . (A) such t(i) f c A , we I- ; t h u s , r f belongs t o f belongs rq A which t o g e t h e r , s i n c e t h e r-topology i s compact convergence. Denote a s X ; then of i s c l o s e d f o r t h e compact open A know by p r o p o s i t i o n 4 . 4 t h a t is a l s o a s . i . d , Frechet space with Xr X i s a l s o i n v a r i a n t by d e r i v a t i o n by and t o a n a t u r a l , s .

We prove by c o n t r a d i c t i o n t h a t t h i s c a s e 5, the always happens. Suppose a l l are d i s t i n c t s . For k . l a r g e enough, v k ( E ) c u t s p ( a ) t E'U -1 For such a f i x e d k , t h e p o i n t s (Ink(a)] a r e a l l d i s t i n c t p 'n,k 'n and t h e r e i s some f E A which s e p a r a t e s them by p r o p o s i t i o n 2 . 4 and by t h e Eaire property of The mappings u A n,m . @a nk(E) n [p(a) o u t s i d e a set v a l u e s on t Z p-'(z) O - f 0 # 0 f(Cm,,) .

Precompact t o p o l o g y ) . S i n c e p o i n t w i s e convergence i m p l i e s uniform convergence on compact 6 ( X I , t h e set 6 ( X I s and 6 ( X I c . I t sets f o r e v e r y eq u i co n t i n u o u s s e t i n of precompact ba la nc e d convex s u b s e t s are e q u a l f o r w i l l b e de note d by SPECTRUM AND MAXIMAL EXTENSIONS 50 K(X). If is complete and any set of is metrizable, 6 ( X I C E K(X) is compact. - [SO, 1 4 1 There i s a locally convex, Hausdorff, topology on 6(X) , denoted by 0 ( X J E such that : a ) i t i s the f i n e s t one t h a t induces on any T € K(X) t h e pointwise convergence, b ) given a c .

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