A characterization of weakly compact sets by convex functions
首发时间:2012-02-14
Abstract:This note introduces a convexity property of convexfunctions defined on a nonempty convex set in Banach spaces and establishes several equivalent characterizations of weakcompactly subsets of Banach spaces by the property. More precisely, let X be a Banach space and let K∈X be a convex bounded subset. Then (i) If X is separable, then K is weakly compact iff thereexists a continuous 2R convex function on K.(ii) If X is nonseparable then K is weakly compact iffthere exists a continuous w2R convex function on K.
keywords: Banach space weakly compact set convex function
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