It is shown, using the Borwein–Preiss variational principle that for every continuous convex function f on a weakly compactly generated space X, every x0 ∈ X and every weakly compact convex symmetric set K such that spanK = X, there is a point of Gâteaux differentiability of f in x0 +K. This extends a Klee’s result for separable spaces. The well-known Mazur’s theorem says that a continuous conv...

where (n1,n2, . . . ,nd)= n∈ Z+. Recently, Thanh [11] proved (1.1) under condition of uniform integrability of {|Xn|p, n∈ Z+}. Mean convergence theorems for sums of random elements Banach-valued are studied by many authors. The reader may refer to Wei and Taylor [12], Adler et al. [2], Rosalsky and Sreehari [9], or more recently, Rosalsky et al. [10], Cabrera and Volodin [3]. However, we are un...

Certain subclasses of B 1 (K), the Baire-1 functions on a compact metric space K, are defined and characterized. Some applications to Banach spaces are given. Let X be a separable infinite dimensional Banach space and let K denote its dual ball, Ba(X *), with the weak* topology. K is compact metric and X may be naturally identified with a closed subspace of C(K). X * * may also be identified wi...

The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of ω1 subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied.

In this paper we examine nonlinear integrodifferential inclusions defined in a separable Banach space. Using a compactness type hypothesis involving the ball measure of noncompactness, we establish two existence results. One involving convex-valued orientor fields and the other nonconvex valued ones.

Klaus Schmidt proved that if a strictly stationary sequence of (say) real-valued random variables is such that the family of distributions of its partial sums is tight, then that sequence is a \coboundary". Here Schmidt's result is extended to some (not necessarily stationary) sequences of random variables taking their values in a separable real Banach space.

We provide a porosity based approach to the differentiability and continuity of real valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone K with non-empty interior. We also show that the set of nowhere K-monotone functions has a σ-porous complement in the space of the continuous functions.

We study the notion of a strongly normal sequence in the dual E∗ of a Banach space E. In particular, we prove that the following three conditions are equivalent: (1) E∗ has a strongly normal sequence, (2) (E∗, σ(E∗, E)) has a Schauder basic sequence, (3) E has an infinite-dimensional separable quotient.

In a recent paper we have shown that most non-expansive Lipschitz functions (in the sense of Baire’s category) have a maximal Clarke subdifferential. In the present paper, we show that in a separable Banach space the set of non-expansive Lipschitz functions with a maximal Clarke subdifferential is not only of generic, but also staunch. 1991 Mathematics Subject Classification: Primary 49J52.