This paper deals with the validity of the Pontryagin duality theorem in the class of metrizable topological groups. We prove that completeness is a necessary condition for the Pontryagin reflexivity of those groups. We also prove that in order for a metrizable separable topological group to be Pontryagin reflexive it is sufficient that the canonical embedding into its bidual group be an algebraic isomorphism.On the other hand, we consider the notion of reflexivity introduced by E. Binz and H. Butzmann. In [5] it was proved that, for topological abelian groups, the Pontryagin reflexivity and the Binz-Butzmann reflexivity, in general, are independent notions. Here we prove that these notions coincide in the class of metrizable topological abelian groups.
We prove that direct and inverse limits of sequences of reflexive Abelian groups that are metrizable or k -spaces, but not necessarily locally compact, are reflexive and dual of each other provided some extra conditions are satisfied by the sequences.
We present a series of examples of precompact, noncompact, reflexive topological Abelian groups. Some of them are pseudocompact or even countably compact, but we show that there exist precompact non-pseudocompact reflexive groups as well. It is also proved that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.
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