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

The free abelian topological group as a subgroup of the free locally convex topological vector space Abstract. The Tkachenko­Uspenskii theorem states that if X is any completely regular Hausdor¤ 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. 1 Introduction and preliminaries Enflo characterized the metric abelian groups G which can be embedded isometrically in a Banach space as those where the invariant metric d has the property that dðg 2 ; eÞ ¼ 2dðg; eÞ for all g in G, where e is the identity element in G. In this paper it is proved that each of the Graev pseudometrics (see [4], [13]) on the free abelian group has this Enflo property. This yields a new proof that for every completely regular Hausdor¤ space X, the free abelian topological group on X is naturally embedded as a topological subgroup of the free http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Group Theory de Gruyter

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

McPhail, Carolyn E.
Journal of Group Theory , Volume 6 (3) – Mar 27, 2003
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References (8)

Publisher
de Gruyter
Copyright
Copyright © 2003 by the
ISSN
1433-5883
eISSN
1435-4446
DOI
10.1515/jgth.2003.027
Publisher site
See Article on Publisher Site

Abstract

Abstract. The Tkachenko­Uspenskii theorem states that if X is any completely regular Hausdor¤ 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. 1 Introduction and preliminaries Enflo characterized the metric abelian groups G which can be embedded isometrically in a Banach space as those where the invariant metric d has the property that dðg 2 ; eÞ ¼ 2dðg; eÞ for all g in G, where e is the identity element in G. In this paper it is proved that each of the Graev pseudometrics (see [4], [13]) on the free abelian group has this Enflo property. This yields a new proof that for every completely regular Hausdor¤ space X, the free abelian topological group on X is naturally embedded as a topological subgroup of the free

Journal

Journal of Group Theory – de Gruyter

Published: Mar 27, 2003

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