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

Cheng, Lixin ; Luo, Zhenghua ; Zhou, Yu

Canadian mathematical bulletin, Tome 56 (2013) no. 2, pp. 272-282

Résumé

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 $X$ .

Détail

DOI : 10.4153/CMB-2011-169-3

Mots-clés : 20M30, 46B10, 46B20, 46E15, 46J10, 49J50, super weakly compact set, dual of normed semigroup, uniform Fréchet differentiability, representation.

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