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Transactions of the American Mathematical Society

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Weak compactness in locally convex spaces


Author: D. G. Tacon
Journal: Trans. Amer. Math. Soc. 180 (1973), 463-474
MSC: Primary 46A25
DOI: https://doi.org/10.1090/S0002-9947-1973-0322467-8
MathSciNet review: 0322467
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Abstract: The notion of weak compactness plays a central role in the theory of locally convex topological vector spaces. However, in the statement of many theorems, completeness of the space, or at least quasi-completeness of the space in the Mackey topology is an important assumption. In this paper we extend the concept of weak compactness in a general way and obtain a number of useful particular cases. If we replace weak compactness by these generalized notions we can drop the completeness assumption from the statement of many theorems; for example, we generalize the classical theorems of Eberlein and Kreĭn. We then consider generalizations of semireflexivity and reflexivity and characterize these properties in terms of our previous ideas as well as in terms of known concepts. In most of the proofs we use techniques of nonstandard analysis.


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Keywords: Locally convex topological vector space, weak compactness, quasi-completeness, semireflexivity and reflexivity, nonstandard analysis
Article copyright: © Copyright 1973 American Mathematical Society