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

**Example text**

Bounded l i n e a r maps a r e i n t r o d u c e d and i m e d i a t e l y used f o r a d e f i n i t i o n o f d i s t r i b u t i o n s ( E x e r c i s e 1- E . 12) The remaining Exercises on t h i s Chapter a r e d e d i c a t e d t o 'von Neumann b o r n o l o g i e s ' , 'bornivorous s e t s ' and b o r n o l o g i c a l convergence f o r f i l t e r s . l A BORNOLOGY on a s e t X i s a f a m i l y @ o f s u b s e t s o f X s a t i s f y i n g t h e following axioms: ( B . I ) : 03 is a covering of X , i .

I): Then: L? is a separated bornoZogicaZ vector space; 39 BORNOLOGICAL CONSTRUCTIONS ( i i ) : For every bounded l i n e a r map u of E i n t o a separated bornological v e c t o r *space G, t h e r e e x i s t s a unique bounded l i n e a r map d of E i n t o G such t h a t : u = iLocp. Proof: ( i ) : This f o l l o w s from t h e f a c t t h a t E l f 0 1 i s s e p a r a t e d ( P r o p o s i t i o n (2) of S e c t i o n 2:11) s i n c e (0) i s b-closed i n E (Proposition ( 1 ) ) . ( i i ) : Since G i s s e p a r a t e d , 0 i s b - c l o s e d i n G ( P r o p o s i t i o n (1) o f S e c t i o n 2:11) hence u-1(0)i s b - c l o s e d i n E &mark (3) o f S e c t i o n 2:11) and, c o n t a i n i n g 0 e E, i t c o n t a i n s {O) a l s o .

I n f a c t , l e t V be a neighbourhood o f zero i n F . S i n c e u i s continuous, W = u - l ( V ) i s a neighbourhood o f zero i n E and hence A C A0 t W, A0 b e i n g a f i n i t e s u b s e t of E. Consequently: u(A) C u(A0) t u(W) C u(A0) t V , 24 BORNOLOGY w i t h u ( A 0 ) a f i n i t e s e t i n F. Thus u ( A ) i s precompact. We now g i v e some f u r t h e r p r o p e r t i e s o f precompact s e t s . ( c ) : I n a topological vector space E t h e closure o f a precompact s e t i s precompact.