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


Published:20110831
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
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 $X$.
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]
