The Glicksberg theorem on weakly compact sets for nuclear groups

Abstract

By the weak topology on an Abelian topological group we mean the topology induced by the family of all continuous characters. A well-known theorem of I. Glicksberg says that weakly compact subsets of locally compact Abelian (LCA) groups are compact. D. Remus and F.J. Trigos-Arrieta [1993. Proceedings Amer. Math. Soc. 117] observed that Glicksberg's theorem remains valid for closed subgroups of any product of LCA groups. Here we show that, in fact, it remains valid for all nuclear groups, a class of Abelian topological groups introduced by the first author in the monograph, “Additive subgroups of topological vector spaces” [1991. Lecture Notes in Math. 1466].