On some new characterizations of weakly compact sets in Banach spaces
Volume 201 / 2010
Studia Mathematica 201 (2010), 155-166
MSC: Primary 46A55, 46B20, 46B50; Secondary 46B03, 58C20, 46G05.
DOI: 10.4064/sm201-2-3
Abstract
We show several characterizations of weakly compact sets in Banach spaces. Given a bounded closed convex set $C$ of a Banach space $X$, the following statements are equivalent: (i) $C$ is weakly compact; (ii) $C$ can be affinely uniformly embedded into a reflexive Banach space; (iii) there exists an equivalent norm on $X$ which has the $w2R$-property on $C$; (iv) there is a continuous and $w^{*}$-lower semicontinuous seminorm $p$ on the dual $X^*$ with $p\geq{\sup_C}$ such that $p^2$ is everywhere Fréchet differentiable in $X^*$; and as a consequence, the space $X$ is a weakly compactly generated space if and only if there exists a continuous and $w^*$-l.s.c. Fréchet smooth (not necessarily equivalent) norm on $X^*$.
