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1. A characterization of weakly compact sets by convex functions | |||
CHENG Qingjin | |||
Mathematics 20 January 2012 | |||
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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. | |||
TO cite this article:CHENG Qingjin. A characterization of weakly compact sets by convex functions[OL].[20 January 2012] http://en.paper.edu.cn/en_releasepaper/content/4462252 |
2. On uniformly convex subsets of Banach spaces | |||
Cheng Qingjin | |||
Mathematics 14 January 2012 | |||
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Abstract:This paper introduces a notion ofuniform convex sets in Banach spaces, which is a localized settingof uniformly convex Banach spaces, and shows that every uniformlyconvex set has many nice properties, such as, every boundeduniformly convex set is weakly compact and admits the Radon-Rieszproperty. It also presents that the metric projection to everynonempty uniformly convex set is always continuous, every convexsubset in a uniformly convex space is uniformly convex and everycompact convex subset in a strictly convex space is also uniformlyconvex. | |||
TO cite this article:Cheng Qingjin. On uniformly convex subsets of Banach spaces[OL].[14 January 2012] http://en.paper.edu.cn/en_releasepaper/content/4462246 |
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