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By H. Hogbe-Nlend

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Extra info for Bornologies and Functional Analysis

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In Functional A n a l y s i s it i s important t o be a b l e t o handle both methods of i n v e s t i g a t i o n . CHAPTER 111 COMPLETE B O R N O L O G IES This Chapter i s devoted t o complete bornologies and complete These spaces a r e t o a r b i t r a r y s e p a r a t e d convex b o r n o l o g i c a l spaces what Banach spaces a r e t o a r b i t rary normed s p a c e s . S i n c e every s e p a r a t e d convex b o r n o l o g i c a l space i s a 'union o f normed s p a c e s ' , a complete convex bornologi c a l space w i l l simply be a 'union o f Banach s p a c e s ' ( i n a p r e cise sense).

Let now Y be t h e q u o t i e n t o f t h e s e t X by an a r b i t r a r y e q u i valence r e l a t i o n , cp denoting t h e canonical map o f X o n t o Y . Then Y , equipped w i t h t h e image bornology ofG3 under cp, i s c a l l e d t h e BORNOLOGICAL QUOTIENT o f (X,@) and t h e bornology cp(a) i s c a l l e d t h e QUOTIENT BORNOLOGY o f @ by t h e given equivalence r e l a t i o n . I f we t a k e f o r X a b o r n o l o g i c a l v e c t o r space ( r e s p . convex b o r n o l o g i c a l space) E and f o r Y t h e q u o t i e n t EIF, where F i s a v e c t o r subspace o f E, t h e n t h e image bornology of (8 i s a v e c t o r ( r e s p .

Then t h e PRODUCT BORNOLOGY on X is the i n i t i a l bornology on X for t h e maps p i . The s e t X, endowed w i t h t h e product bornology, i s c a l l e d t h e the Sets ( X i @ < ) . f o r a l l i e I . n: i d Proof: A s e t o f t h e form B = n: B i i s c l e a r l y bounded f o r t h e i€I product bornology, s i n c e i t s p r o j e c t i o n s a r e bounded. Conversely, i f A i s a s u b s e t o f X which i s bounded f o r t h e product bornology, 31 BORNOLOGICA L CONSTRUCTIONS l e t A i = p i ( A ) f o r every i e I, and B = A;.

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