On Mackey topology for groups

The present paper is a contribution to fill in a gap existing between the t heory of topological vector spaces and that of topological abelian groups. Topological vector spaces have been extensively studied as part of Function al Analysis. It is natural to expect that some important and elegant theore ms about topological vector spaces may have analogous versions for abelian topological groups. The main obstruction to get such versions is probably t he lack of the notion of convexity in the framework of groups. However, the introduction of quasi-convex sets and locally quasi-convex groups by Vilen kin [26] and the work of Banaszczyk [1] have paved the way to obtain theore ms of this nature. We study here the group topologies compatible with a giv en duality. We have obtained, among others, the following result: for a com plete metrizable topological abelian group, there always exists a finest lo cally quasi-convex topology with the same set of continuous characters as t he original topology. We also give a description of this topology as an G-t opology and we prove that, for the additive group of a complete metrizable topological vector space, it coincides with the ordinary Mackey topology.