The free abelian topological group as a subgroup of the free locally convex topological vector space

摘要

The Tkachenko-Uspenskii theorem states that if X is any completely regular Hausdorff space then the free abelian topological group on X is embedded naturally in the additive topological group of the free locally convex topological vector space on X. A new and simple proof of that result is presented in this paper, using Enflo's characterization of those metric abelian groups which can be embedded isometrically as a subgroup of a Banach space. En route it is shown that each of the Graev pseudometrics on a free abelian group has the Enflo property.