Václav Zizler

NORMS WITH LOCALLY LIPSCHITZIAN DERIVATIVES

If a separable Banach spaceX admits a real valued function ф with bounded nonempty support, φ 艂 is locally Lipschitzian and if no subspace ofX is isomorphic toco, thenX admits an equivalent twice Gateaux differentiable norm whose first Frechet differential is Lipschitzian on the unit sphere ofX.

Smoothness in weakly compactly generated Banach spaces

Received August 25, 1982; revised April 15, 1983 If X* is a weakly compactly generated (WCG) Banach space, then X admits an equivalent C’smooth norm. If a WCG Banach space X admits a Ck-smooth function with bounded support, then X admits Ck-smooth partitions of unity.

Smoothness and its equivalents in weakly compactly generated Banach spaces

Abstract Some equivalent properties of the existence of Frechet smooth norms in weakly compactly generated (WCG) Banach spaces are shown. Heredity of WCG property in such spaces is proved. The property of having a shrinking Markusevic basis is hereditary in all Banach spaces. Projections in WCG spaces and their duals are studied.

Farthest points in W *-compact sets

and call r(x) the farthest distance from x to C. Equivalently, r(x) is the radius of the smallest ball of centre x , containing C. The function r is convex as supremum of such functions, and continuous since \r(x) — r(y)\ ^ ||x — y\\, for all x , y £ X . A point z £ C is called a farthest point of C if there exists x £ X such that ||x — z|| = r(x). The existence of a farthest point of C is equivalent to the fact that the set

The structure of uniformly Gâteaux smooth Banach spaces

It is shown that a Banach spaceX admits an equivalent uniformly Gâteaux smooth norm if and only if the dual ball ofX* in its weak star topology is a uniform Eberlein compact.

Hilbert-generated spaces☆

We classify several classes of the subspaces of Banach spaces X for which there is a bounded linear operator from a Hilbert space onto a dense subset in X. Dually, we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in Hilbert spaces or in c0(Γ) spaces in their weak topology. The existence of such embeddings is characterized by the existence of certain uniformly Gâteaux smooth norms.

A renorming of dual spaces

If Banach spacesX,X* are both weakly compactly generated, thenX has an equivalent norm whose dual onX* is locally uniformly rotund.

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).

The non-linear geometry of Banach spaces after Nigel Kalton

This is a survey of some of the results which were obtained in the last twelve years on the non-linear geometry of Banach spaces. We focus on the contribution of the late Nigel Kalton.

Weakly compact sets and smooth norms in Banach spaces

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.