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# On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate

Published:2011-08-31
Printed: Jun 2013
• Lixin Cheng,
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
• Zhenghua Luo,
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
• Yu Zhou,
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
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## Abstract

In this note, we first give a characterization of super weakly compact convex sets of a Banach space $X$: a closed bounded convex set $K\subset X$ is super weakly compact if and only if there exists a $w^*$ lower semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly Fréchet differentiable on each bounded set of $X^*$. Then we present a representation theorem for the dual of the semigroup $\textrm{swcc}(X)$ consisting of all the nonempty super weakly compact convex sets of the space $X$.
 Keywords: super weakly compact set, dual of normed semigroup, uniform Fréchet differentiability, representation
 MSC Classifications: 20M30 - Representation of semigroups; actions of semigroups on sets 46B10 - Duality and reflexivity [See also 46A25] 46B20 - Geometry and structure of normed linear spaces 46E15 - Banach spaces of continuous, differentiable or analytic functions 46J10 - Banach algebras of continuous functions, function algebras [See also 46E25] 49J50 - Frechet and Gateaux differentiability [See also 46G05, 58C20]

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