A weakly Stegall space that is not a Stegall space
by
Sivajah Somasundaram
Department of Mathematics, The University of Waikato, Hamilton, New Zealand
Coauthors: Dr. Warren B. Moors (Department of Mathematics, The University of Waikato, Hamilton, New Zealand)

We say that a Banach space X is weak Asplund (almost weak Asplund) if every continuous convex function defined on a non-empty open convex subset A of X is Gâteaux differentiable at the points of a residual (everywhere second category) subset of A. A set-valued mapping j:X --> 2^Y acting between topological spaces X and Y is called an usco mapping if for each x in X, j(x) is a non-empty compact subset of Y and for each open set W in Y, {x in X:j(x) subset or equal W} is open in X. An usco mapping j:X --> 2^Y is called minimal if its graph does not properly contain the graph of any other usco defined on X. In the study of weak Asplund spaces, one can consider the following class of topological spaces which are defined in terms of minimal uscos. We say that a topological space X belongs to the class of Stegall (weakly Stegall) spaces if for every Baire (complete metric) space B and minimal usco mapping j:B --> 2^X, j is single-valued at the points of a residual (everywhere second category) subset of B. It is known that a Banach space X is weak Asplund (almost weak Asplund) if (X^*, weak^*) lies in the class of Stegall (weakly Stegall) spaces. In this talk we show that there is a Banach space X whose dual, equipped with the weak^* topology, is in the class of weakly Stegall spaces but not in the class of Stegall spaces. By contrast, it is still open as to whether there is an almost weak Asplund space that is not a weak Asplund space.

Date received: October 14, 2001