Bounded duality in topological abelian groups

Abstract

We define the β-duality for topological Abelian groups by means of the notion of Hejcman of boundedness in uniform spaces. A real locally convex space considered as an Abelian topological group is β-reflexive iff it is reflexive in the ordinary sense for locally convex spaces. Thus, β-reflexivity is the natural extension to Abelian topological groups of the well-known notion of reflexivity. We prove: 1) A locally compact Abelian group is β-reflexive. 2) A β-reflexive metrizable group is reflexive in Pontryagin sense. 3) The β-bidual of a metrizable group is also a metrizable group.