Weak compactness in locally convex spaces

Tacon, D. G.

doi:10.1090/S0002-9947-1973-0322467-8

Weak compactness in locally convex spaces
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by D. G. Tacon

Trans. Amer. Math. Soc. 180 (1973), 463-474

DOI: https://doi.org/10.1090/S0002-9947-1973-0322467-8

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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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Topologische linear Räume

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