On some new characterizations of weakly compact sets in Banach spaces

Lixin Cheng, Qingjin Cheng, and Zhenghua Luo. "On some new characterizations of weakly compact sets in Banach spaces." Studia Mathematica 201.2 (2010): 155-166. <http://eudml.org/doc/286674>.

@article{LixinCheng2010,
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 $p ≥ sup_\{C\}$ 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*.},
author = {Lixin Cheng, Qingjin Cheng, Zhenghua Luo},
journal = {Studia Mathematica},
keywords = {weakly compact sets; renorming; smoothness; convexity; WCG space; Banach space},
language = {eng},
number = {2},
pages = {155-166},
title = {On some new characterizations of weakly compact sets in Banach spaces},
url = {http://eudml.org/doc/286674},
volume = {201},
year = {2010},
}

TY - JOUR
AU - Lixin Cheng
AU - Qingjin Cheng
AU - Zhenghua Luo
TI - On some new characterizations of weakly compact sets in Banach spaces
JO - Studia Mathematica
PY - 2010
VL - 201
IS - 2
SP - 155
EP - 166
AB - 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 $p ≥ sup_{C}$ 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*.
LA - eng
KW - weakly compact sets; renorming; smoothness; convexity; WCG space; Banach space
UR - http://eudml.org/doc/286674
ER -