J. Aust. Math. Soc.
75 (2003), 69-83
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Banach-Dieudonné theorem revisited
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Elena Martín-Peinador
Dept. de Geometría y Topología
Universidad Complutense de Madrid
Spain
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).
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