Metric duality for Abelian groups

TL;DR

This paper introduces metric duality for topological Abelian groups, extending classical duality concepts, and shows that all locally compact Abelian groups can be equipped with reflexive proper metrics.

Contribution

It extends duality theory to topological Abelian groups by defining metric duality and demonstrates the existence of reflexive metrics for all LCA groups.

Findings

Every Polish LCA group admits a reflexive proper metric.

All LCA groups possess reflexive (proper) metric structures.

The concept generalizes classical duality for normed spaces.

Abstract

The main aim of the paper is to introduce the concept of metric duality in the category of topological Abelian groups that extends the classical notion of duality for normed vector spaces and behaves quite nicely for LCA groups (equipped with nice metrics). In particular, it is shown that each Polish LCA group admits a reflexive proper metric and, more generally, all LCA groups possess reflexive (proper) metric structures.