Bornologies and Functional Analysis: Introductory course on by Henri Hogbe-Nlend (Eds.)

By Henri Hogbe-Nlend (Eds.)

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Such a bornology i s c a l l e d t h e COMPACT BORNOLOGY of X . The compact bornology o f a s e p a r a t e d t o p o l o g i c a l v e c t o r space i s a v e c t o r bornology. I n f a c t , l e t us denote t h i s bornology b y a . For every s c a l a r A e M , t h e map x -+ Ax o f E i n t o E i s continuous, hence f o r every compact s e t A C E, AA i s compact. S i m i l a r l y , t h e c o n t i n u i t y o f t h e map ( x , y ) x t y o f E x E i n t o E ensures t h a t t h e s e t A t B i s compact whenever A and B a r e compact s u b s e t s o f E.

REMARK (2): Note t h a t a n element o f t h e b - c l o s u r e o f a s e t A C E i s n o t , i n g e n e r a l , t h e b o r n o l o g i c a l l i m i t of a sequence o f p o i n t s o f A even i f A i s a subspace o f E ( s e e E x e r c i s e 2 - E . 8 ) . On t h i s m a t t e r , see a l s o E x e r c i s e 1 - E . l l concerning t h e b o r n o l o g i c a l con- vergence o f f i l t e r s . - PROPOSITION (2) : Let E be a bornoZogicaZ v e c t o r space, { O } . t h e b-closure o f 101 i n E, 8 the q u o t i e n t E/{O} and cp:E -+ E t h e canonical map.

The f a m i l y k of equicontinuous subsets o f L ( E , F ) i s a vector borno logy on L ( E,F 1. T h i s borno logy i s cal led t h e EQUICONTINUOUS BORNOLOGY of L ( E , F ) and i s a convex bornology if F i s l o c a l l y convex. We s h a l l now prove t h i s a s s e r t i o n . S i n c e every element o f L ( E , F ) i s continuous by d e f i n i t i o n , R covers L ( E , F ) and i s a l s o c l e a r l y h e r e d i t a r y . Let us show t h a t i s s t a b l e under v e c t o r a d d i t i o n .

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