Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T14:37:17.090Z Has data issue: false hasContentIssue false
  • English
  • Français

FRÉCHET INTERMEDIATE DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS ON ASPLUND SPACES

Part of: Existence theories

Published online by Cambridge University Press:  13 March 2009

J. R. GILES
J. R. GILES*
Affiliation:
University of Newcastle, NSW 2308, Australia (email: John.Giles@newcastle.edu.au)
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.

The deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

Keywords

Lipschitz functionsFréchet differentiabilityAsplund and Asplund generated spaces

MSC classification

Secondary: 49J52: Nonsmooth analysis
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 309 - 317
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Fabian, M., Loewen, P. and Wang, X., ‘ϵ-Fréchet differentiability of Lipschitz functions and applications’, J. Convex Anal. 13 (2006), 695–709.Google Scholar
[2] Fabian, M. and Preiss, D., ‘On intermediate differentiability of Lipschitz functions on certain Banach spaces’, Proc. Amer. Math. Soc. 113 (1991), 733–740.CrossRefGoogle Scholar
[3] Giles, J. R. and Sciffer, S., ‘A generic differentiability property of Lipschitz functions on Asplund spaces’, J. Nonlinear Convex Anal. 3 (2002), 353–363.Google Scholar
[4] Giles, J.R. and Sciffer, S., ‘Generalising generic differentiability properties from convex to locally Lipschitz functions’, J. Math. Anal. Appl. 188 (1994), 833–854.CrossRefGoogle Scholar
[5] Lindenstrauss, J. and Preiss, D., ‘A new proof of Fréchet differentiability of Lipschitz functions’, J. Eur. Math. Soc. 2 (2000), 199–216.CrossRefGoogle Scholar
[6] Phelps, R. R., ‘Dentability and extreme points in Banach spaces’, J. Funct. Anal. 16 (1974), 78–90.CrossRefGoogle Scholar
[7] Phelps, R. R., Convex Functions, Monotone Operators and Differentiability, 2nd edn, Lecture Notes in Mathematics 1364 (Springer, Berlin, 1992).Google Scholar
[8] Preiss, D., ‘Differentiability of Lipschitz functions on Banach spaces’, J. Funct. Anal. 91 (1990), 312–345.Google Scholar