By G. Coeuré (Eds.)

ISBN-10: 0080871224

ISBN-13: 9780080871226

ISBN-10: 0444106219

ISBN-13: 9780444106216

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

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U ( Theorem 6 . 1 3 . - The mcCimzl extension [Q(X)] sp" [@ (X)€ ] , whenever E is metrizabZe. - Suppose E a Frechet c. v . ned i n sp [&(XI,] . 9, 8 (X)€ is contained in 8. induces on topology which i s t h e compact topology s i n c e sp* [o(%),] ; there- by t h e previous theorem. Then x [o ( X ) ] E' o p is contai- t h e precompact i s complete. Then w e have E sp 6 (XI, . - which a r e u s e f u l . The main Comment There a r e o t h e r t o p o l o g i e s on one o f them is t h e Nachbin topology [62] c a r r i e r s of f u n c t i o n a l s on @ ( X I [18,29] 6' ( X ) which i s used t o study t h e .

A l l Te K(X) a b a s i s of a f i l t e r and f o r any M sets a a l s o be longs i n t h e r e s t r i c t i o n of 0 i s a neighbourhood o f a . To prove i s t h e f i n e s t o n e, w e have only b ) t o c he c k, t h u s g i v e n a b a l a n c e d convex open s e t V in G , then @-'(V) be longs t o 7 and the p r o o f is complete. The t o p o l o g y by . Let T€ be g i ve n and K(X) AT b e t h e spa c e spanned and normed by t h e Minkowski-norm a s s o c i a t e d w i t h T mapping AT -).

Since O(XL nuous. 8 the quotient space E' . E is a semi-Monte1 space and is finitely dimensional by Riesz's theorem, (iii) implies (i), since is finitely dimensional. Then the extension mrp u* from Q ~ x ) , (resp. 6 1 onto b t ~ ) ,(resp. O ( Y L I is topological whenever E is metriaable. 10 the b-topologies are bornological and we have to verify that the range of any bounded set by u* and ( ~ ~ 1 -is l bounded for compact open topology. This property is obvious for (u*)-' , Let T be a bounded set in b(X)(= , so T is contained in a suitable s .

### Analytic Functions and Manifolds in Infinite Dimensional Spaces by G. Coeuré (Eds.)

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