On some new characterizations of weakly compact sets in Banach spaces

Lixin Cheng ¹   ; Qingjin Cheng ¹   ; Zhenghua Luo ¹

¹ School of Mathematical Sciences, Xiamen University Xiamen, 361005, China

Studia Mathematica, Tome 201 (2010) no. 2, pp. 155-166

Résumé

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 $X^*$.

Détail

DOI : 10.4064/sm201-2-3

Keywords: several characterizations weakly compact sets banach spaces given bounded closed convex set banach space following statements equivalent weakly compact affinely uniformly embedded reflexive banach space iii there exists equivalent norm which has r property there continuous * lower semicontinuous seminorm dual * geq sup everywhere chet differentiable * consequence space weakly compactly generated space only there exists continuous * l chet smooth necessarily equivalent norm nbsp *