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Semicontinuous functions and convex sets in C(K) spaces

Part of: Linear function spaces and their duals Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

J. P. Moreno
J. P. Moreno
Affiliation:
Dpto. Mateáticas Facultad de Ciencias Universidad Autónoma de MadridMadrid 28049Spain e-mail: josepedro.moreno@uam.es
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Abstract

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The stability properties of the family ℳ of all intersections of closed balls are investigated in spaces C(K), where K is an arbitrary Hausdorff compact space. We prove that ℳ is stable under Minkowski addition if and only if K is extremally disconnected. In contrast to this, we show that ℳ is always ball stable in these spaces. Finally, we present a Banach space (indeed a subspace of C[0, 1]) which fails to be ball stable, answering an open question. Our results rest on the study of semicontinuous functions in Hausdorff compact spaces.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 1 , February 2007 , pp. 111 - 121
Copyright
Copyright © Australian Mathematical Society 2007

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