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 such that p² 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*.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm201-2-3,
author = {Lixin Cheng and Qingjin Cheng and Zhenghua Luo},
title = {On some new characterizations of weakly compact sets in Banach spaces},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {155-166},
zbl = {1217.46013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm201-2-3}
}
Lixin Cheng; Qingjin Cheng; Zhenghua Luo. On some new characterizations of weakly compact sets in Banach spaces. Studia Mathematica, Tome 196 (2010) pp. 155-166. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm201-2-3/