On Schwartz Groups

Abstract

In this paper we introduce a notion of a Schwartz group, which turns out to be coherent with the well known concept of a Schwartz topo- logical vector space. We establish several basic properties of Schwartz groups and show that free topological Abelian groups, as well as free locally convex spaces, over a hemicompact k{space are Schwartz groups. We also prove that every hemicompact k{space topological group, in particular the Pontryagin dual of a metrizable topological group, is a Schwartz group.