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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 super weakly compact set, dual of normed semigroup, uniform Fréchet differentiability, representation
MSC Classifications: 20M30, 46B10, 46B20, 46E15, 46J10, 49J50 show english descriptions Representation of semigroups; actions of semigroups on sets
Duality and reflexivity [See also 46A25]
Geometry and structure of normed linear spaces
Banach spaces of continuous, differentiable or analytic functions
Banach algebras of continuous functions, function algebras [See also 46E25]
Frechet and Gateaux differentiability [See also 46G05, 58C20]
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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