Marián Fabian

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.

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.

Sequential normal compactness versus topological normal compactness in variational analysis

Abstract We study relationships between two normal compactness properties of sets in Banach spaces that play an essential role in many aspects of variational analysis and its applications, particularly in calculus rules for generalized differentiation, necessary optimality and suboptimality conditions for optimization problems, etc. Both properties automatically hold in finite-dimensional spaces and reveal principal features of the infinite-dimensional variational theory. Similar formulations of these properties involve the weak ∗ convergence of sequences and nets, respectively, containing generalized normal cones in duals to Banach spaces. We prove that these properties agree for a large class of Banach spaces that include weakly compactly generated spaces. We also show that they are always different in Banach spaces whose unit dual ball is not weak ∗ sequentially compact. Moreover, the sequential and topological normal compactness properties may not coincide even in non-separable Asplund spaces that admit an equivalent C∞-smooth norm.

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.

Nonsmooth Characterizations of Asplund Spaces and Smooth Variational Principles

We show that the Asplund property of Banach spaces is not only sufficient but also a necessary condition for the fulfillment of some basic results in nonsmooth analysis involving Fréchet-like normals and subdifferentials as well as their sequential limits. In this way we obtain new characterizations of Asplund spaces within the framework of nonsmooth analysis. Then we study several versions of smooth variational principles in Asplund spaces, provide necessary and sufficient conditions for the validity of such principles, and establish their relationships with certain subdifferential properties of lower semicontinuous functions.

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.

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

A real-valued function f defined on a Banach space X is said to be intermediately differentiable at x E X if there is 4 E X* such that for every h E X the value (4, h) lies between the upper and lower derivatives of f at x in the direction h . We show that if Y contains a dense continuous linear image of an Asplund space and X is a subspace of Y, then every locally Lipschitz function on X is generically intermediately differentiable. Let (X, 11 * 11) be a Banach space with dual X* and duality pairing between X* and X denoted by (., *). Recall that the upper and lower derivatives of a function f: X -R at x E X in a direction h E X are defined by D+f(x, h) = lim sup t[f(x + th) f(x)] tlO and D+f(x, h) = liminf l[f(x, th) f(x)] respectively. The function f is said to be intermediatly differentiable at x E X, with intermediate derivative 4 E X*, if D+f(x,h)>(4,h)>D+(x,h) forallheX. The aim of this note is to prove the following statement. Theorem. Suppose that a Banach space Y contains a dense continuous linear image of an Asplund space and that X is a subspace of Y. Then every locally Lipschitz function defined on an open subset Q of X is intermediately differentiable at every point of a residual subset of Q. The Banach spaces containing a dense continuous linear image of an Asplund space have been extensively studied by Ch. Stegall [11], who calls them GSG spaces. Among other things, he also proved [11, ?4, Remark 3] that the nonw.c.g. subspace of some LI (,u), ,u finite, constructed by H. Rosenthal [10] is Received by the editors April 23, 1990 and, in revised form, July 11, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 46G05; Secondary 46B22, 58C20.

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.

On intermediate differentiability of Lipschitz functions on certain Banach spaces

The paper presents a general primal space classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several primal space derivative-like objects – slopes – are used to characterize the error bound property of extended-real-valued functions on metric sapces.