Locally convex topologies and the convex compactness property

Author:

Ostling E. G.,Wilansky A.

Abstract

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.

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Cited by 5 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Krein’s Theorem in the Context of Topological Abelian Groups;Axioms;2022-05-12

2. Group duality with the topology of precompact convergence;Journal of Mathematical Analysis and Applications;2005-03

3. Banach-Dieudonné Theorem Revisited;Journal of the Australian Mathematical Society;2003-08

4. On Weak Compactness in Biprojective Tensor Product Spaces;Mathematische Nachrichten;1977

5. Completeness and intertwined completeness of locally convex spaces;Mathematical Proceedings of the Cambridge Philosophical Society;1977-01

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