Some More Recent Results concerning Weak Asplund Spaces

In this paper, we will provide some examples of Banach spaces that are Gâteaux differentiability spaces but not weak Asplund, weak Asplund but not in class(
̃), in class(
̃) but whose dual space is not weak∗ fragmentable. We begin with some definitions. A Banach space X is called a weak Asplund space [almost weak Asplund] (Gâteaux differentiability space) if each continuous convex function defined on it is Gâteaux differentiable at the points of a residual [everywhere second category] (dense) subset. While it is easy to see that every weak Asplund space is an almostweak Asplund space and every almost weak Asplund space is a Gâteaux differentiability space, it has been a long standing question as to whether there are in fact Gâteaux differentiability spaces that are not weak Asplund. In the study of weak Asplund spaces, several classes of topological spaces have played a prominent role; two of which we describe below. A set-valued mapping φ : X → 2Y acting between topological spaces X and Y is called an usco mapping if for each x ∈ X , φ(x) is a nonempty compact subset of Y and for each open set W in Y , {x ∈ X : φ(x) ⊆W} is open in X . An usco mapping φ : X → 2Y is called a minimal usco if its graph does not contain, as a proper subset, the graph of any other usco defined on X . Below we recall some of the basic properties of minimal uscos. Proposition 1.1 [2, Proposition 3.1.2]. Let φ : X → 2Y be an usco acting between topological spaces X and Y . Then φ is a minimal usco if and only if, for each pair of open subsets U of X andW of Y with φ(U)∩W = ∅, there exists a nonempty open subset V of U such that φ(V)⊆W .