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**Example text**

F i n a l l y , f o r e v e r y compact A C E, t h e c i r c l e d h u l l o f A i s compact, s i n c e i t i s t h e image o f t h e s e t D x A ( D t h e compact u n i t b a l l o f M ) under t h e continuous map (X,x> Ax o f M x E i n t o E . In g e n e r a l , t h e compact bornology o f of a t o p o l o g i c a l v e c t o r s p a c e , even a normed one, i s n o t convex (cf. E x e r c i s e 4 - E . 9 ; s e e , however, Example (10) below). For t h i s r e a s o n one o f t e n c o n s i d e r s t h e following bornology: EXAMPLE (5) : -f -+ The Bornology of Compact D i s k s of a TopoZogical Vector Space: A compact d i s k i n a s e p a r a t e d topologi c a l v e c t o r space E i s a s e t w h i c h i s s i m u l t a n e o u s l y compact and d i s k e d .

Thus U i ( x ) 4 0 . + + + Conversely, i f t h e c o n d i t i o n o f t h e P r o p o s i t i o n i s s a t i s f i e d and M i s a bounded v e c t o r subspace o f E , t h e n u i ( M ) i s a bounded subspace o f E i f o r every i e I . Since E i i s s e p a r a t e d , u i ( M ) reduces t o {O) and hence M c o n t a i n s no non-zero v e c t o r s . COROLLARY: (a) : Every product of separated bornological vector spaces is separated; (b) : Every bornological subspace of a separated bornological vector space is separated; (c) : Every i n t e r s e c t i o n of separated vector bornologies i s separated; (d) : Every p r o j e c t i v e l i m i t of separated bornological v e c t o r spaces is separated.

REMARK ( 1 ) : Let Y be a b o r n o l o g i c a l s e t and l e t X be endowed w i t h X is t h e i n i t i a l bornology f o r t h e maps u i . Then a map u : Y bounded if and only i f U i O U i s bounded f o r every i E I. -f 2 ~ 1 . 1 Base of an I n i t i a l Bornology With t h e n o t a t i o n o f Theorem (1) , l e t : ui -1 (ai) = Then t h e family020 = IUi -1 (A): AeaiI n ui-l(Gi) f o r every i e I. i s a b a s e o f t h e i n i t i a l born- ie l ology (B on X f o r t h e maps u i . 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 .