By Everitt

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C ) : I n a topological vector space E t h e closure o f a precompact s e t i s precompact. This follows from t h e f a c t t h a t E has a base o f c l o s e d neighbourhoods o f z e r o . ( d ) : I n a separated l o c a l l y convex space t h e precompact bornology i s convex. We s h a l l show d i r e c t l y t h a t t h e d i s k e d h u l l o f a precompact s e t is precompact. We b e g i n by showing t h a t t h e disked h u l l o f a f i n i t e s e t i s precompact. Let { a l , . ,a,} be a f i n i t e s u b s e t o f E, l e t C be i t s disked h u l l and l e t B = { ( X I , n .

OF CONVEX BORNOLOGICAL S P A C E S ) . -+ 2 ~ 8 ' 2 I n d u c t i v e L i m i t Eornologies Let ( X i , V j i ) be an i n d u c t i v e system o f b o r n o l o g i c a l s e t s ( r e s p . b o r n o l o g i c a l v e c t o r s p a c e s , r e s p . convex b o r n o l o g i c a l spaces) and 34 FUNDAMENTAL l e t X be t h e s e t ( r e s p . l). For every i e l , denote by t h e bornology o f X i and by V i t h e canonical map o f X i i n t o x. The INDUCTIVE L I M I T BORNOLOGY on x w i t h r e s p e c t t o bornologies @ii s t h e f i n a l bornology on X for.

2) : i s hereditary under i n c l u s i o n , i . e . if A E(B and B i s a subset of X contained i n A , then B e @ ; (B i s s t a b l e under f i n i t e union. A p a i r (X,@) c o n s i s t i n g o f a s e t X and a bornology 6 on X i s (B. 3 ) : (B c a l l e d a BORNOLOGICAL S E T , and t h e elements o f 6 a r e c a l l e d t h e BOUNDED S U B S E T S O f A BASE O F A BORNOLOGY 6 on X i s any subfamily of such x. t h a t every element o f (72 i s contained i n an element o f G3o. A fami l y @ o f s u b s e t s o f X i s a base f o r a bornology on X i f and o n l y i f 030 covers X and every f i n i t e union o f elements o f 630 i s cont a i n e d i n a member o f 6 0 .

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