Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T07:46:06.711Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Bishop, E. and Phelps, R. R., ‘A proof that every Banach space is subreflexive’, Bull. Amer. Math. Soc. 67 (1961), 97–98.CrossRefGoogle Scholar
[2]Giles, J. R., Kenderov, P. S., Moors, W. B. and Sciffer, S. D., ‘Generic differentiability of convex functions on the dual of a Banach space’, Pacific J. Math. 172 (1996), 413–431.CrossRefGoogle Scholar
[3]Giles, J. R. and Moors, W. B., ‘A continuity property related to Kuratowski's index of non- compactness, its relevance to the drop property and its implications for differentiability theory’, J. Math. Anal. Appl. 178 (1993), 247–268.CrossRefGoogle Scholar
[4]Haydon, R., ‘Locally uniformly convex norms on Banach spaces and their duals’, in preparation.Google Scholar
[5]Jayne, J. E., Namioka, I. and Rogers, C. A., ‘σ-fragmentable Banach spaces’, Mathematika 39 (1992), 161–188.CrossRefGoogle Scholar
[6]Kenderov, P. S. and Giles, J. R., ‘On the structure of Banach spaces with Mazur's intersection property’, Math. Ann. 291 (1991), 463–473.CrossRefGoogle Scholar
[7]Kenderov, P. S. and Moors, W. B., ‘Game characterization of fragmentability of topological spaces’, in: Proceedings of the 25th Spring Conf. Union of Bulg. Mathematicians, Kazanlak, 1996), Math. and Education in Math., pp. 8–18.Google Scholar
[8]Kenderov, P. S., Moors, W. B. and Revalski, J. P., ‘Dense continuity and selections of set-valued mappings’, Serdica Math. J. 24 (1998), 49–72.Google Scholar
[9]Kenderov, P. S., Moors, W. B. and Sciffer, S. D., ‘Norm attaining functionals on C(T)’, Proc. Amer. Math. Soc. 126 (1998), 153–157.CrossRefGoogle Scholar
[10]Matejdes, M., ‘Quelques remarques sur la quasi-continuité des multifonctions’, Math. Slovaca 37 (1987), 267–271.Google Scholar
[11]Moltó, A., Orihuela, J., Troyanski, S. and Valdivia, M., ‘On weakly locally uniformly rotund Banach spaces’, J. Funct. Anal. 163 (1999), 252–271.CrossRefGoogle Scholar
[12]Moors, W. B., ‘A Banach space whose dual norm is locally uniformly rotund is a generic continuity space’, unpublished manuscript, 1994.Google Scholar
[13]Moors, W. B., ‘The relationship between Goldstine's theorem and the convex point of continuity property’, J. Math. Anal. Appl. 188 (1994), 819–832.CrossRefGoogle Scholar
[14]Moors, W. B. and Giles, J. R., ‘Generic continuity of minimal set-valued mappings’, J. Austral. Math. Soc. (Series A) 63 (1997), 238–268.CrossRefGoogle Scholar
[15]Namioka, I. and Pol, R., ‘Mappings of Baire spaces into function spaces and Kadeč renormings’, Israel J. Math. 78 (1992), 1–20.CrossRefGoogle Scholar
[16]Negrepontis, S., Banach spaces and topology. Handbook of set theoretic topology (North-Holland, Amsterdam, 1984).Google Scholar
[17]Oxtoby, J. C., Measure and category. A survey of the analogies between topological and measure spaces (Springer, New York, 1971).Google Scholar
[18]Raja, M., ‘Kadeč norms and Borel sets in a Banach space’, Studia Math. 136 (1999), 1–16.CrossRefGoogle Scholar
[19]Troyanski, S., ‘On a property of the norm which is close to local uniform rotundity’, Math. Ann. 271 (1985), 305–314.CrossRefGoogle Scholar
[20]Troyanski, S., ‘On some generalisations of denting points’, Israel J. Math. 88 (1994), 175–188.CrossRefGoogle Scholar