CARMA COLLOQUIUM Speaker: Prof John Giles, School of Mathematical and Physical Sciences, The University of Newcastle Title: On a Weakly Uniformly Rotund Dual of a Banach Space Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 4:00 pm, Thu, 26th Jul 2012 Abstract: Hajek proved that a WUR Banach space is an Asplund space. This result suggests that the WUR property might have interesting consequences as a dual property. We show that (i) every Banach Space with separable second dual can be equivalently renormed to have WUR dual, (ii) under certain embedding conditions a Banach space with WUR dual is reflexive. [Permanent link] CARMA SEMINAR Speaker: Prof John Giles, School of Mathematical and Physical Sciences, The University of Newcastle Title: Persistence properties for Banach spaces Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 4:00 pm, Thu, 8th Sep 2011 Abstract: We are interested in local geometrical properties of a Banach space which are preserved under natural embeddings in all even dual spaces. An example of this behaviour which we generalise is: if the norm of the space $X$ is Fréchet differentiable at $x \in S(X)$ then the norm of the second dual $X^{**}$ is Fréchet differentiable at $\hat{x}\in S(X)$ and of $X^{****}$ at $\hat{\hat{x}} \in S(X^{****})$ and so on.... The results come from a study of Hausdorff upper semicontinuity properties of the duality mapping characterising general differentiability conditions satisfied by the norm. [Permanent link] 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 Abstract: 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. [Permanent link]