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By Henri Hogbe-Nlend (Eds.)

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Additional resources for Bornologies and Functional Analysis: Introductory course on the theory of duality topology-bornology and its use in functional analysis

Sample text

EXAMPLE (2) : The Bomology Defined by a Family of Semi-Norms: Let E be a v e c t o r space and r = ( p i ) i E 1 a family of semi-norms on E indexed by a non-empty s e t I. We a g r e e t o say t h a t a s u b s e t A Of E i S a SUBSET BOUNDED FOR THE FAMILY r OF SEMI-NORMS i f f o r every i e I , pi(A) i s bounded i n m . 'I s e p a r a t e d i f and o n l y i f r s e p a r a t e s E, i . e . i f f o r every x e E , CL" 0 , t h e r e e x i s t s i e l such t h a t p i ( x:) 0 . This Example w i l l be g e n e r a l i s e d i n Chapter I1 t o t h e n o t i o n of ' i n i t i a l b o r n o l o g y ' .

In f a c t , on t h e one hand, every i s e v i d e n t l y bounded f o r G , s i n c e u i ( A ) e(Ri f o r element A o f each i e l . On t h e o t h e r hand, i f A e a , t h e n , f o r e v e r y i e I , u;(A) i s bounded i n X i and hence t h e r e e x i s t s B i e a t such t h a t u i ( A ) C B i , i . e . A C u i - l ( B i ) . Thus t h e i n t e r s e c t i o n o f t h e s e t s ui-'(Bi) belongs t o 60and c o n t a i n s A , and t h e a s s e r t i o n f o l l o w s . The most important p a r t i c u l a r c a s e s o f i n i t i a l b o r n o l o g i e s are given i n t h e f o l l o w i n g S e c t i o n s 2:2-5.

Note t h a t t h e r e always e x i s t s a bornology which c o n t a i n s A, namely t h e bornology U3 = 6 ( X ) whose bounded s e t s a r e a l l t h e subs e t s o f X and, i f X i s a v e c t o r s p a c e , t h i s bornology i s convex. 5 FUNDAMENTAL BORNOLOGICAL PROJECTIVE LIMITS 2:5'1 Bornological P r o j e c t i v e Systems Let I be a non-empty d i r e c t e d s e t and l e t ( X i y U i j ) be a p r o j e c t i v e system o f s e t s , indexed by I ( c f . S e c t i o n 0 * A . 2 ) , such t h a t f o r every i e I , X i i s a b o r n o l o g i c a l s e t with bornology The system ( x i , u i j ) i s c a l l e d a P R O J E C T I V E S Y S T E M O F BORNOLOGICAL S E T S i f t h e maps U i j : X j + X i a r e bounded whenever i 6 j.

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