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Banach-Dieudonné Theorem Revisited

Part of: Topological and differentiable algebraic systems Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

Montserrat Bruguera  and
Elena Martín-Peinador
Montserrat Bruguera
Affiliation:
Dept. de Matemática Aplicada I Universidad Politécnica de CataluñaSpain e-mail: m.montserrat.bruguera@upc.es
Elena Martín-Peinador
Affiliation:
Dept. de Geometría y Topología Universidad Complutense de MadridSpain e-mail: peinador@mat.ucm.es
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Abstract

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We prove that in the character group of an abelian topological group, the topology associated (in a standard way) to the continuous convergence structure is the finest of all those which induce the topology of simple convergence on the corresponding equicontinuous subsets. If the starting group is furthermore metrizable (or even almost metrizable), we obtain that such a topology coincides with the compact-open topology. This result constitutes a generalization of the theorem of Banach-Dieudonné, which is well known in the theory of locally convex spaces.

We also characterize completeness, in the class of locally quasi-convex metrizable groups, by means of a property which we have called the quasi-convex compactness property, or briefly qcp (Section 3).

Keywords

Completemetrizable groupcontinuous convergence structureequicontinuous weak* topologycompact-open topologyBanach-Dieudonné theoremdual grouplocally quasi-convex group.

MSC classification

Secondary: 22A05: Structure of general topological groups 46A04: Locally convex Fréchet spaces and (DF)-spaces 46A19: Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than ${bf R}$, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 1 , August 2003 , pp. 69 - 83
Copyright
Copyright © Australian Mathematical Society 2003

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