CONTENTS Introduction..................................................................................................................................................5 Chapter 1. Uniformly rotund directions..................................................................................................7 Chapter 2. Duality properties...................................................................................................................15 Chapter 4. Some remarks to the theory of weakly uniformly rotund spaces..................................21 Chapter 5. An application to a fixed point theory..................................................................................27 Chapter 6. On one Mazur's theorem.......................................................................................................28 References..................................................................................................................................................32
σ-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 σ-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.
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