V. Zizler

A quantitative version of Krein's Theorem

A quantitative version of Kreins Theorem on convex hulls of weak compact sets is proved. Some applications to weakly compactly gen- erated Banach spaces are given.

On Weak Compactness in

We will use the concept of strong generating and a simple renorming theorem to give new proofs to slight generalizations of some results of Argyros and Rosenthal on weakly compact sets in L1(μ) spaces for finite measures μ. The purpose of this note is to show that a simple transfer renorming theorem explains why L1(μ)-spaces, for finite measures μ, share some properties with superreflexive spaces, though there is no one-to-one bounded linear operator from L1(μ) into any reflexive space if L1(μ) is nonseparable [19, p. 232]. ∗Supported by grants IAA 100190610 and AVOZ 101 905 03 (Czech Republic). †Supported in part by Project MTM2005-08210 (Spain), the Generalitat Valenciana and the Universidad Politécnica de Valencia (Spain). ‡Supported by grants IAA 100 190 502 and AVOZ 101190503 (Czech Republic).

L_1

Preface.- Basic linear structure.- Basic linear geometry.- Biorthogonal systems.- Smoothness, smooth approximation.- Nonlinear geometry.- Some more nonseparable problems.- Some applications.- Bibliography.- List of concepts and problems.- Symbol index.- Subject index.

Spaces

We collect facts about Gul’ko, descriptive, Gruenhage, and fragmentable compact spaces. In several instances we provide direct proofs of the results discussed. We show how they reflect the geometrical structure of corresponding Banach spaces C(K). In particular, we provide proofs, by a simple transfer of Day’s norm, of recent renorming results for the duals C(K)* due to M. Raja and to R. Smith.ResumenSe recogen resultados acerca de espacios compactos de Gul’ko, descriptivos de Gruenhage y fragmentables. En ciertos casos se proporcionan pruebas directas de los resultados tratados. Se muestra cómo reflejan la estructura geométrica de los correspondientes espacios C(K). En particular, se proporcionan pruebas, mediante una simple transferencia de la norma de Day, de recientes resultados de renormamiento para los espacios duales C(K)* debidos a M. Raja y R. Smith.

Open problems in the geometry and analysis of banach spaces

We study countable splitting of Markushevich bases in weakly Lindelof Banach spaces in connection with the geometry of these spaces.