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

Abstract

AbstractIn this note, we first give a characterization of super weakly compact convex sets of a Banach space X: a closed bounded convex set K ⊂ X is super weakly compact if and only if there exists a w* lower semicontinuous seminorm p with p ≥ σK ≌ supxєK 〈.,x〉 such that p2 is uniformly Fréchet differentiable on each bounded set of X*. Then we present a representation theoremfor the dual of the semigroup swcc(X) consisting of all the nonempty super weakly compact convex sets of the space X.