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Compactness in Hom(G, H)

Published online by Cambridge University Press:  20 November 2018

H. H. Corson  and
I. Glicksberg
H. H. Corson
Affiliation:
University of Washington, Seattle, Washington
I. Glicksberg
Affiliation:
University of Washington, Seattle, Washington
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Let G be a locally compact abelian group with Bohr compactification Ga. Then [3, Theorem 1.2] any subset F of G compact in Ga is necessarily compact in G; alternatively, any closed non-compact subset F of G has its closure F– in Ga ≠ F; hence F –\F ≠ ø. One of our aims in the present note is to give a result (Corollary 6) which asserts that F –\F has no points which are Gδs, so that F–\F is a perfect set. Another aim is to give an extension of a cited result of [3] in which commutativity and local compactness are essentially irrelevant, and to unify the proofs.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 164 - 170
Copyright
Copyright © Canadian Mathematical Society 1970

References

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2. Day, M. M., Normed linear spaces (Springer, Berlin, 1958).Google Scholar
3. Glicksberg, I., Uniform boundedness for groups, Can. J. Math. 14 (1962), 269–276.Google Scholar
4. Grothendieck, A., Critères de compacité dans les espaces fonctionelles généraux, Amer. J. Math. 74 (1952), 168–186.Google Scholar