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The Glicksberg Theorem on Weakly Compact Sets for Nuclear Groups
Author(s) -
BANASZCZYK W.,
MARTÍNPEINADOR E.
Publication year - 1996
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1996.tb36794.x
Subject(s) - abelian group , topological group , locally compact space , mathematics , topology (electrical circuits) , product (mathematics) , class (philosophy) , pure mathematics , discrete mathematics , combinatorics , computer science , geometry , artificial intelligence
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 ].