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A dual differentiation space without an equivalent locally uniformly rotund norm

Part of: Normed linear spaces and Banach spaces; Banach lattices Calculus on manifolds; nonlinear operators

Published online by Cambridge University Press:  09 April 2009

Petar S. Kenderov  and
Warren B. Moors
Petar S. Kenderov
Affiliation:
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria e-mail: kenderov@math.bas.bg
Warren B. Moors
Affiliation:
Department of Mathematics, The University of Auckland, Auckland, New Zealand e-mail: moors@math.auckland.ac.nz
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Abstract

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A Banach space (X, ∥ · ∥) is said to be a dual differentiation space if every continuous convex function defined on a non-empty open convex subset A of X* that possesses weak* continuous subgradients at the points of a residual subset of A is Fréchet differentiable on a dense subset of A. In this paper we show that if we assume the continuum hypothesis then there exists a dual differentiation space that does not admit an equivalent locally uniformly rotund norm.

Keywords

locally uniformly rotund normBishop-Phelps setRadon-Nikodým property

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 58C20: Differentiation theory (Gateaux, Fréchet, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 3 , December 2004 , pp. 357 - 364
Copyright
Copyright © Australian Mathematical Society 2004

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