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Banach-Dieudonné Theorem Revisited

Published online by Cambridge University Press:  09 April 2009

Montserrat Bruguera
Affiliation:
Dept. de Matemática Aplicada I Universidad Politécnica de CataluñaSpain e-mail: [email protected]
Elena Martín-Peinador
Affiliation:
Dept. de Geometría y Topología Universidad Complutense de MadridSpain e-mail: [email protected]
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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).

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2003

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