Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-10-04T00:13:47.295Z Has data issue: false hasContentIssue false

A Weak Hadamard Smooth Renorming of L1(Ω, μ)

Published online by Cambridge University Press:  20 November 2018

Jonathan M. Borwein
Affiliation:
Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario N2L 3G1
Simon Fitzpatrick
Affiliation:
Department of Mathematics University of Western Australia Nedlands, W.A. 6019 Australia
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.

We show that L1(μ) has a weak Hadamard differential)le renorm (i.e. differentiable away from the origin uniformly on all weakly compact sets) if and only if μ is sigma finite. As a consequence several powerful recent differentiability theorems apply to subspaces of L1.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1993

References

[Boll Borwein, J. M., Minimal cuscos and subgradients of Lipschitz functions. In: Fixed Point Theory and its Applications, (eds. J.-B. Bäillon and M. Thera), Pitman Lecture Notes in Math., Longman, Essex, 1991, 5782.Google Scholar
[Bo2] , Asplund spaces are “sequentially reflexive ”, University of Waterloo, July, 1991, Research Report CORR, 91–14.Google Scholar
[BF] Borwein, J. M. and Fabian, M., On convex functions having points of Gateaux differentiability which are not points of Fréchet differentiability, Can. J. Math., in press.Google Scholar
[BP] Borwein, J. M. andPreiss, D., A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303(1987), 517527.Google Scholar
[Da] Mahlon Day, M., Normed Linear Spaces, 3rd éd., Springer-Verlag, New York, 1973.Google Scholar
[DFJP] Davis, W. J., Figiel, T., Johnson, W. B. and Pelczynski, A., Factoring weakly compact operators, J. Functional Analysis 17(1974), 311327.Google Scholar
[DGZ] R. Deville, Godefroy, G. and Zizler, V., Un principe variational utilisant des fonctions bosses, C. R. Acad. Sci. Paris, (1991), 281286.Google Scholar
[Dil] Diestel, J., Geometry ofBanach spaces, Lecture notes in Math. (485), Springer-Verlag, Berlin, 1975.Google Scholar
[Di2] Diestel, J., Sequences and series in Banach spaces, Graduate texts in Math. (92), Springer-Verlag, Berlin, 1984.Google Scholar
[Gi] Giles, J., Convex analysis with application in differentiation of convex functions, Research notes in Math. (58), Pitman, London, 1982.Google Scholar
[La] Lacey, H., The isometric theory of classical Banach spaces, Springer-Verlag, Berlin, 1974.Google Scholar
[LT] Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I: sequence spaces, Springer-Verlag, Berlin, 1979.Google Scholar
[NP] Namioka, I. and Phelps, R., Banach spaces which are Asplund spaces, Duke Math. J. 42(1975), 735750.Google Scholar
[Or] Orno, P., On J. Borwein's concept of sequentially reflexive Banach spaces, 1991, preprint.Google Scholar
[Ph] Phelps, R., Convex functions, monotone operators, and differentiability, Lecture notes in Math. (1364), Springer-Verlag, Berlin, 1989.Google Scholar
[Pr] Preiss, D., Differentiability of Lipschitz functions on Banach spaces, J. Functional Analysis 91(1990), 312345.Google Scholar
[PPN] Preiss, D., Phelps, R. R. and Namioka, I., Smooth Banach spaces, weak Asplund spaces, and monotone or USCO mappings, Israel J. Math. 72(1990), 257279.Google Scholar
[St] Stegall, C., Gateaux differentiation of functions on a certain class of Banach spaces. In: Functional Analysis: Surveys and Recent Results III, North Holland, 1984,3546.Google Scholar