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Sequential Compactness of X Implies a Completeness Property for C(X)

Published online by Cambridge University Press:  20 November 2018

M. Rajagopalan  and
R. F. Wheeler
M. Rajagopalan
Affiliation:
Memphis State University, Memphis, Tennessee
R. F. Wheeler
Affiliation:
Memphis State University, Memphis, Tennessee
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A locally convex Hausdorff topological vector space is said to be quasicomplete if closed bounded subsets of the space are complete, and von Neumann complete if closed totally bounded subsets are complete (equivalently, compact). Clearly quasi-completeness implies von Neumann completeness, and the converse holds in, for example, metrizable locally convex spaces. In this note we obtain a class of locally convex spaces for which the converse fails. Specifically, let X be a completely regular Hausdorff space, and let CC(X) denote the space of continuous real-valued functions on X, endowed with the compact-open topology.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 1 , 01 February 1976 , pp. 207 - 210
Copyright
Copyright © Canadian Mathematical Society 1976

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