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ArticleOriginal scientific text

Title

On Mackey topology for groups

Authors 1, 2, 3

Affiliations

  1. Facultad de Ciencias, Universidad de Navarra, 31080 Pamplona, Spain
  2. Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
  3. Muskhelishvili Institute of Comp. Math., Georgian Academy of Sciences, Tbilisi 93, Georgia

Abstract

The present paper is a contribution to fill in a gap existing between the theory of topological vector spaces and that of topological abelian groups. Topological vector spaces have been extensively studied as part of Functional Analysis. It is natural to expect that some important and elegant theorems about topological vector spaces may have analogous versions for abelian topological groups. The main obstruction to get such versions is probably the lack of the notion of convexity in the framework of groups. However, the introduction of quasi-convex sets and locally quasi-convex groups by Vilenkin [26] and the work of Banaszczyk [1] have paved the way to obtain theorems of this nature. We study here the group topologies compatible with a given duality. We have obtained, among others, the following result: for a complete metrizable topological abelian group, there always exists a finest locally quasi-convex topology with the same set of continuous characters as the original topology. We also give a description of this topology as an S-topology and we prove that, for the additive group of a complete metrizable topological vector space, it coincides with the ordinary Mackey topology.

Keywords

ENG
locally convex space, Mackey topology, continuous character, weakly compact, locally quasi-convex group

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Studia Mathematica
1999/132/3
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Pages:
257-284
Main language of publication
English
Accepted
1997-09-25
Published
1999
Exact and natural sciences