- CARMA COLLOQUIUM
- Speaker: Prof John Giles, School of Mathematical and Physical Sciences, The University of Newcastle
- Title: A Continuity Characterisation of Asplund Spaces
- Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
- Time and Date: 4:00 pm, Thu, 2nd Sep 2010
An Asplund space is a Banach space which possesses desirable differentiability properties enjoyed by Euclidean spaces. Many characterisations of such spaces fall into two classes: (i) those where an equivalent norm possesses a particular general property, (ii) those where every equivalent norm possesses a particular property at some points of the space. For example: (i) X is an Asplund space if there exists an equivalent norm Frechet differentiable on the unit sphere of the space, (ii) X is an Asplund space if every equivalent norm is Frechet differentiable at some point of its unit sphere. In 1993 (F-P) showed that (i) X is an Asplund space if there exists an equivalent norm strongly subdifferentiable on the unit sphere of the space and in 1995 (G-M-Z) showed that (ii) X separable is an Asplund space if every equivalent norm is strongly subdifferentiable at a nonzero point of X. Problem: Is this last result true for non-separable spaces? In 1994 (C-P) showed (i) X is an Asplund space if there exists an equivalent norm with subdifferential mapping Hausdorff weak upper semicontinuous on its unit sphere. We show: (ii) X is an Asplund space if every continuous gauge on X has a point where its subdifferential mapping is Hausdorff weak upper semicontinuous with weakly compact image which is some way towards solving the problem.
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