Vicente Montesinos

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.

A CHARACTERIZATION OF SUBSPACES OF WEAKLY COMPACTLY GENERATED BANACH SPACES

We prove that a Banach space X is a subspace of a weakly compactly generated Banach space if and only if, for every e > 0, X can be covered by a countable collection of bounded closed convex symmetric sets the weak∗ closure in X∗∗ of each of them lies within the distance e from X. As a corollary, we give a new, short functional-analytic proof to the known result that a continuous image of an Eberlein compact is Eberlein. ∗Supported by grants AV 1019003, 1019301, and GACR 201/01/1198 (Czech Republic). †Supported in part by Project pb96-0758 (Spain), Project BFM2002-01423 and by the Universidad Politecnica de Valencia, ‡Supported by NSERC 7926 (Canada).

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

Two smoothness characterisations of weakly compact sets in Banach spaces are given. One that involves pointwise lower semicontinuous norms and one that involves projectional resolutions of identity.

Spaces

Given any infinite cardinal τ, there exists no Banach space of density r, which is Asplund or has the Point of Continuity Property and is universal for all reflexive spaces of density τ.

Weakly compact sets and smooth norms in Banach spaces

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.

Universality of Asplund spaces

An extreme point of the closed unit ball of a Banach space is said to be preserved if it is extreme of the closed unit ball of the bidual space; otherwise it is called unpreserved. The beginning of the present work takes the form of a survey on this topic, presenting some elementary facta about those concepts—usually with new proofs—and discussing in particular Katznelson’s solution to a Phelps’ question on preserved extreme points, not available, to our knowledge, in the literature. In a second part, some new results are presented. Since some of them depend on the concept of polyhedrality, we first review several results on this topic. Then we present Godun renorming theorem for the class of nonreflexive Banach spaces, and Morris renorming result—with a new proof—on separable Banach spaces containing a copy of (c_0). We show that, under some extra conditions—polyhedrality—a similar renorming, this time adding smoothness, can be defined ensuring strict convexity with all points in the unit sphere unpreserved extreme. We finalize this work by presenting what—to our knowledge—is the first nonseparable result of this kind for the natural class of the weakly compactly generated Banach spaces.

Open problems in the geometry and analysis of banach spaces

A close connection of the strict convexity of the Day norm to the concept of the Gruenhage compacta is shown. As a byproduct we give an elementary characterization of Gul’ko compacta in the sigma-product of lines and a more elementary proof of Mercourakis’ renorming result for Vasak spaces. This note is a result of our effort to classify those Banach spaces with dual ball Corson (in its weak∗ topology) that would admit a Gâteaux smooth renorming. Given a non-empty set S, let Σ(S) := {x ∈ [−1, 1] ; support of x is countable}. We shall always assume that Σ(S) is endowed with its product topology. A compact space K is called a Corson compact if K is homeomorphic to a subset of Σ(S). It was proved in [AM, p. 425] that a Banach space X with dual ball Corson need not in general admit any equivalent Gâteaux differentiable norm. We find here a sufficient condition for a Gâteaux smooth renorming that uses the strict convexity of the Day norm on the dual space and is closely related to the notion of Gruenhage compacta (cf. [Gru], [AM, p. 424], [Ri, Def 2.1]). As a corollary, we prove a renorming theorem which gives, in particular, a result in [Me]. The result in this note is related to the result of Raja that X∗ admits a dual norm that is weak∗ locally uniformly rotund if and only if BX∗ in its weak∗ topology is a descriptive compact [Ra]. As a byproduct of our efforts, we obtain a characterization, in the Sokolov’s style [S], of Gul’ko compacta lying in the space Σ(S) in the spirit of the characterization of Eberlein compacta given in [Fa]. ∗Supported by grants AV 1019003, A 1019301 and GACR 201/01/1198. †Supported in part by Project BFM2002-01423 and a grant of the Universidad Politecnica de Valencia. This author thanks the Department of Mathematical and Statistical Sciences of the University of Alberta, Edmonton, Canada, and the Institute of Mathematics of the Czech Academy of Sciences for their support and hospitality. ‡Supported by grant NSERC 7926.

On Preserved and Unpreserved Extreme Points

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.

The day norm and Gruenhage compacta

We characterize the duals and biduals of the