Library of Congress Cataloging-in-Publieation Data Funetional analysis andinfinite-dimensional geometry / Mariän Fabian . . . [et al.). p. cm . -(CMS books in mathematies ; 8) Ineludes bibliographical references andindex. I. Funetional analysis. 2. Banaeh spaces. I. Fabian, Mariän J. 1I. Series. QA320 .F793 2001 5I5'.7--de2 I 00-053773

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The paper presents a general classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several derivative-like objects both from the primal as well as from the dual space are used to characterize the error bound property of extended-real-valued functions on a Banach space.

A quantitative version of Krein's Theorem on convex hulls of weak compact sets is proved. Some applications to weakly compactly generated Banach spaces are given.

We give characterizations of weakly compactly generated spaces, their subspaces, Vašák spaces, weakly Lindelöf determined spaces as well as several other classes of Banach spaces related to uniform Gâteaux smoothness, in terms of the presence of a total subset of the space with some additional properties. In addition, we describe geometrically, when possible, these classes by means of suitable smoothness or rotundity of the norm. As a consequence, we get new, functional analytic proofs of several theorems on (uniform) Eberlein, Gul'ko and Talagrand compacta.