On Mackey topology for groups

M. Chasco; E. Martín-Peinador; V. Tarieladze

Studia Mathematica (1999)

  • Volume: 132, Issue: 3, page 257-284
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Chasco, M., Martín-Peinador, E., and Tarieladze, V.. "On Mackey topology for groups." Studia Mathematica 132.3 (1999): 257-284. <http://eudml.org/doc/216599>.

@article{Chasco1999,
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.},
author = {Chasco, M., Martín-Peinador, E., Tarieladze, V.},
journal = {Studia Mathematica},
keywords = {locally convex space; Mackey topology; continuous character; weakly compact; locally quasi-convex group; group topologies compatible with a given duality; complete metrizable topological Abelian group; locally quasiconvex topology},
language = {eng},
number = {3},
pages = {257-284},
title = {On Mackey topology for groups},
url = {http://eudml.org/doc/216599},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Chasco, M.
AU - Martín-Peinador, E.
AU - Tarieladze, V.
TI - On Mackey topology for groups
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 3
SP - 257
EP - 284
AB - 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.
LA - eng
KW - locally convex space; Mackey topology; continuous character; weakly compact; locally quasi-convex group; group topologies compatible with a given duality; complete metrizable topological Abelian group; locally quasiconvex topology
UR - http://eudml.org/doc/216599
ER -

References

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