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Krein-Smulian-Type Theorems

Part of: Probability theory on algebraic and topological structures Topological linear spaces and related structures Classical measure theory

Published online by Cambridge University Press:  24 October 2014

Surjit Singh Khurana
Surjit Singh Khurana*
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA, (khurana@math.uiowa.edu)
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Abstract

Let (E, ℱ) be a weakly compactly generated Frechet space and let ℱ0 be another weaker Hausdorff locally convex topology on E. Let X be an ℱ-bounded compact subset of (E, ℱ0). The ℱ0-closed convex hull of X in E is then ℱ0-compact. We also give a new proof, without using Riemann–Lebesgue-integrable (Birkoff-integrable) functions, with the result that if (E, ∥ · ∥) is any Banach space and ℱ0 is fragmented by ∥ · ∥, then the same result holds. Furthermore, the closure of the convex hull of X in ℱ0-topology and in the original topology of E is the same.

Keywords

equicontinuous setsweak compactnessbarycentres

MSC classification

Primary: 46A04: Locally convex Fréchet spaces and (DF)-spaces 46A50: Compactness in topological linear spaces; angelic spaces, etc. 60B05: Probability measures on topological spaces
Secondary: 46A55: Convex sets in topological linear spaces; Choquet theory 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 2 , June 2015 , pp. 441 - 444
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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References

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