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A duality theory for Banach spaces with the Convex Point-of-Continuity Property

University of British Columbia

A norm ||⋅|| on a Banach space X is Fréchet differentiable at x ∈ X if there is a functional ∫∈ X* such that [Formula Omitted] This concept reflects the smoothness characteristics of X. A dual Banach space X* has the Radon-Nikodym Property (RNP) if whenever C ⊂ X* is weak*-compact and convex, and ∈ > 0, there is an x ∈ X and an ⍺ > 0 such that diameter [Formula Omitted] this property reflects the convexity characteristics of X*. Culminating several years of work by many researchers, the following theorem established a strong connection between the smoothness of X and the convexity of X*: Every equivalent norm on X is Fréchet differentiable on a dense set if and only if X* has the RNP. A more general measure of convexity has been recently receiving a great deal of attention: A dual Banach space X* has the weak* Convex Point-of-Continuity Property (C*PCP) if whenever ɸ ≠ C ⊂ X* is weak*-compact and convex, and ∈ > 0, there is a weak*-open set V such that V ⋂ C ≠ ɸ and diam V ⋂ C < ∈. In this thesis, we develop the corresponding smoothness properties of X which are dual to C*PCP. For this, a new type of differentiability, called cofinite Fréchet differentiability, is introduced, and we establish the following theorem: Every equivalent norm on X is cofinitely Fréchet differentiable everywhere if and only if X* has the C*PCP. Representing joint work with R. Deville, G. Godefroy and V. Zizler, an alternate approach is developed in the case when X is separable. We show that if X is separable, then every equivalent norm on X which has a strictly convex dual is Fréchet differentiable on a dense set if and only if X* has the C*PCP, if and only if every equivalent norm on X which is Gâteaux differentiable (everywhere) is Fréchet differentiable on a dense set. This result is used to show that if X* does not have the C*PCP, then there is a subspace Y of X such that neither Y* nor (X/Y)* have the C*PCP, yet both Y and X/Y have finite dimensional Schauder decompositions. The corresponding result for spaces X* failing the RNP remains open.

Text

2010-08-13

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http://hdl.handle.net/2429/27313

Mathematics

University of British Columbia

Graduate