Weak compactness and $\sigma $-Asplund generated Banach spaces
Volume 181 / 2007
Studia Mathematica 181 (2007), 125-152
MSC: 46B20, 46B50.
DOI: 10.4064/sm181-2-2
Abstract
$\sigma$-Asplund generated Banach spaces are used to give new characterizations of subspaces of weakly compactly generated spaces and to prove some results on Radon–Nikodým compacta. We show, typically, that in the framework of weakly Lindelöf determined Banach spaces, subspaces of weakly compactly generated spaces are the same as $\sigma$-Asplund generated spaces. For this purpose, we study relationships between quantitative versions of Asplund property, dentability, differentiability, and of weak compactness in Banach spaces. As a consequence, we provide a functional-analytic proof of a result of Arvanitakis: A compact space is Eberlein if (and only if) it is simultaneously Corson and quasi-Radon–Nikodým.
