Abstract

Let X and Y be two convex spaces over the same (real or complex) field F. We consider a general method of defining convex topologies on the dual of a convex space, taking as neighborhoods of the origin the polars of certain sets in the convex space. Here we have proved that any finite sum of compact sets is compact. Also the sum of a compact and a closed set in a convex space is closed. Let V be the vector space of all continuous linear mappings of XintoY. Let A be any bounded subsets of X and B a base of absolutely convex neighborhoods in Y. Define WA,B= {t: t (A) B} for each A A and B B. Then WA,Bis absolutely convex and absorbent. The topology is then called the topology of A – convergence. If X is a barreled space, then any point wise bounded set of continuous linear mappings of X into Y isequicontinuous.

References

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