By Henri Hogbe-Nlend
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Extra resources for Nuclear and Conuclear Spaces
Is isomorphjc b t o a quotient of a d i r e c t s u m ) of a f a m i l y of m e m b e r s of V ( E ) e a c h b having d i m e n s i o n c d i m E . c be the s e t of all c . b. s . f ( E ) which h a v e d i m e n s i o n b s d i m E a n d , a s l i n e a r s p a c e s , a r e s u b s e t s of a fixed v e c t o r s p a c e E Next, l e t 0 It follows f r o m above that a n y m e m b e r of V ( E ) 0 b is i s o m o r p h i c t o a quotient of a d i r e c t s u m of m e m b e r s of C. T h e with dim E > d i m E.
S Remark ( 5 ) . compact and hence the same i s true of a bounded l i n e a r map of c into E , by Proposition ( 3 ) of Section 1 : 1 . Finally, we leave it to the r e a d e r t o supply the s i m p l e proof of the following PROPOSITION ( 2 ) . - The class o f a l l Silva spaces (resp. FrGchet - Schwartz s p a c e s ) i s closed under the operations of taking : (i) quotient s p a c e s , (ii) closed s u b s p a c e s , (iii) isomorphic i m a g e s and (iv) countable d i r e c t s u m s ( r e s p .
S . Proof. - (resp. 1 . c . s . ) - ( r e s p . rt) of a l l infra -Schwartz b is a bornological ( r e s p . topological) u l t r a - v a r i e t y . The c l a s s e 5 The proofs a r e essentially the s a m e a s f o r T h e o r e m ( 3 ) and . i t s corollary and s o we l i m i t o u r s e l v e s to showing, a s a n example, that if E <% and F is a closed subspace of E , be a bounded subset of E/F and l e t fl : E then t h e r e e x i s t s a bounded s e t A in E E Let B b' be the quotient m a p ; then E / F 6 3 E/F such that B cfl ( A ) .