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Locally convex topologies and the convex compactness property

Published online by Cambridge University Press:  24 October 2008

E. G. Ostling  and
A. Wilansky
E. G. Ostling
Affiliation:
Hofstra, Lehigh, and Reading Universities
A. Wilansky
Affiliation:
Hofstra, Lehigh, and Reading Universities
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Extract

1. Introduction. A locally convex space is said to have the convex compactness property (sometimes abbreviated to cc) if the absolutely convex closure of each compact set is compact. This important property is the subject of Krein's theorem (3) 24.5(4′). It is strictly weaker than bounded completeness and can sometimes be substituted for that assumption; for example, a useful result, related to the Banach–Mackey theorem, says that in a space with cc, all admissible topologies have the same bounded sets (5). As another example, it is well known that if X is bornological, X′ is strongly complete (see (1), theoreml); but if X has cc as well, we can strengthen this result to conclude, (2) 19C, that X′ is complete with its Mackey topology, indeed with the topology Ta (using the notation of section 3), where T is the original topology of X.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 1 , January 1974 , pp. 45 - 50
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

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