Bornologies and Functional Analysis by H. Hogbe-Nlend

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By H. Hogbe-Nlend

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Sufficiency: Suppose t h a t ( 0 1 i s b - c l o s e d i n E and t h a t (Xn> i s a sequence having l i m i t s x and y i n E. The sequence x n - x n = 0 converges t o x - y , hence x - y = 0 and E i s s e p a r a t e d (Proposi t i o n (4) o f S e c t i o n 1 : 4 ) . We a r e now ready t o g i v e t h e c r i t e r i o n f o r t h e s e p a r a t i o n o f bornological quotients. a bornologica2 v e c t o r s p a c e and l e t M be a subspace of E. The q u o t i e n t EIM i s s e p a r a t e d if and o n l y if M i s b o r n o l o g i c a l l y c l o s e d i n E.

Hence, i f we i d e n t i f y E and Id1), we can c o n s i d e r on E t h e d i r e c t sum bornology w i t h r e s p e c t t o copies o f t h e canonical bornology o f M and t h e space E t h e n becomes t h e b o r n o l o g i c a l i n d u c t i v e l i m i t of i t s f i n i t e - d i m e n s i o n a l subspaces M n , n em. T h i s bornology i s c a l l e d t h e FINITE-DIMENSIONAL BORNOLOGY on E . I t i s t h e f i n e s t v e c t o r bornology on E and i s always convex. 2:9’5 B o r n o l o g i c a l l y Complementary Subspaces Let E be a b o r n o l o g i c a l v e c t o r space and l e t M,N be subspaces o f E such t h a t E i s t h e a l g e b r a i c d i r e c t sum of M and N .

2 PROPOSITION (1): Let E be a bornoZogica1 vector space and l e t (xn) be a sequence i n E . The following a s s e r t i o n s are equiva Zent : (i) : The sequence (5,) converges bornologicaZly t o 0 ; ( i i ) : There e x i s t s a circZed bounded s e t B C E and a decreasing sequence (an) of p o s i t i v e r e a l numbers, tending t o 0 , such t h a t x n e a n B f o r every n e m ; 26 BORNOLOGY c E such t h a t , given any E > 0, we can f i n d an i n t e g e r N ( E ) f o r which xn e EB whenever n 2 N ( E ) .

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