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
