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

Published online by Cambridge University Press:  20 November 2018

Lixin Cheng ,
Zhenghua Luo  and
Yu Zhou
Lixin Cheng
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China e-mail: lxcheng@xmu.edu.cnluozhenghua@hotmail.com roczhoufly@126.comroczhoufly@126.com
Zhenghua Luo
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China e-mail: lxcheng@xmu.edu.cnluozhenghua@hotmail.com roczhoufly@126.comroczhoufly@126.com
Yu Zhou
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China e-mail: lxcheng@xmu.edu.cnluozhenghua@hotmail.com roczhoufly@126.comroczhoufly@126.com
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Abstract

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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\,\ge \,{{\sigma }_{K}}\,\equiv \,{{\sup }_{x\in K}}\left\langle \,\cdot \,,\,x \right\rangle $ such that ${{P}^{2}}$ is uniformly Fréchet differentiable on each bounded set of ${{X}^{*}}$. Then we present a representation theoremfor the dual of the semigroup swcc$\left( X \right)$ consisting of all the nonempty super weakly compact convex sets of the space $X$.

Keywords

20M3046B1046B2046E1546J1049J50super weakly compact setdual of normed semigroupuniform Fréchet differentiabilityrepresentation.
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 2 , 01 June 2013 , pp. 272 - 282
Copyright
Copyright © Canadian Mathematical Society 2013

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