J. Aust. Math. Soc.  75 (2003), 69-83
Banach-Dieudonné theorem revisited

Montserrat Bruguera
  Dept. de Matemática Aplicada I
  Universidad Politécnica de Catalu\ na
Elena Martín-Peinador
  Dept. de Geometría y Topología
  Universidad Complutense de Madrid

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).
Download the article in PDF format (size 122 Kb)

TeXAdel Scientific Publishing ©  Australian MS