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On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate |
Canad. Math. Bull. 56(2013), 272-282
Published:2011-08-31
Printed: Jun 2013
Printed: Jun 2013
- Lixin Cheng,
- Zhenghua Luo,
- Yu Zhou,
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] |