# Summer School on Topological Vector Spaces by L. Waelbroeck By L. Waelbroeck

Lawsuits 1972

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I~ L. n If I-)oO - To show that a subset of that it absorbs every set B 53- E is bornivorous, it is sufficient to show belonging to the collection ~ of all bounded, balanced, convex sets which contain such that 2 ~B B n B . For every B E ~ o The balanced, convex hull R of L C~ S. 1) R ~ L C S. 1). The balanced, convex hull S! ~ L = S i=l z=A--*~c(i~i = O. For large n we have S! 1). 5. Theorem (Dieudonn~). A finlte-eodimensional subspace of a bornological space is bornological. 6. Lemma (Valdivia ).

4. Let ~ the vector space and V be a collection of absorbing, balanced, convex subsets of E. 5. , or equivalently • Two different collec- 10 The collection of generate the locally convex . A seml-normon E is a map p from E into the set ~+ positive real numbers which satisfies p(x+y) & p(x) + p(y) and =}klp(x) The closed semi-ball (x%p(x)~ for all 0 satisfies (I) - (4) and so defines a locally can generate the same locally convex structure. structure defined by ~ A> which contain a finite intersection of sets of W~, E, where x,y6E, %6~\$.

Thus, given a collection ~D of seml-nor~s on E, the closed semi- balls (or equivalently the open semi-balls) pertaining to the a locally convex structure on E. Conversely, every locally convex structure can be so generated since the 5auge ("Minkowskl functional") = Inf { ~ ! ~ p E ~ D generate 0, x ~ ~ v } of an absorbing, balanced, o o n w x Pv(X) set V is seml-norm. 6. space. Exmmple. Let A Nachbin family continuous functions on v~ X Zf with ~f X be a completely regular (Hausdorff) topological on X is a collection of positive, upper semi- such that for m a x ( ~ v l ( x ) , ~ v 2 ( x ) ) ~ v(x) , vector space of all continuous functions for all = su Vl,V 2 6 ~ and v 6 ~.