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SEPARATION OF CONVEX SETS IN EXTENDED NORMED SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices Topological linear spaces and related structures General convexity

Published online by Cambridge University Press:  26 February 2015

G. BEER  and
J. VANDERWERFF
G. BEER
Affiliation:
Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, CA 90032, USA email [email protected]
J. VANDERWERFF*
Affiliation:
Department of Mathematics, La Sierra University, 4500 Riverwalk Parkway, Riverside, CA 92515, USA email [email protected]
*
✉[email protected]
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Abstract

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We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$, that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

Keywords

extended normed spaceconvex setcontinuous separationproper separationstrict separationstrong separationcoreconvex cone

MSC classification

Primary: 46A55: Convex sets in topological linear spaces; Choquet theory
Secondary: 46B20: Geometry and structure of normed linear spaces 52A05: Convex sets without dimension restrictions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 2 , October 2015 , pp. 145 - 165
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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