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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Differentiability of cone-monotone functions on separable Banach space


Authors: Jonathan M. Borwein, James V. Burke and Adrian S. Lewis
Journal: Proc. Amer. Math. Soc. 132 (2004), 1067-1076
MSC (2000): Primary 26B25; Secondary 90C29
Published electronically: July 14, 2003
MathSciNet review: 2045422
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Abstract: Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gâteaux differentiability of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone with non-empty interior. This seemingly arduous restriction is useful, since it covers the case of directionally Lipschitz functions, and necessary. We show by way of example that most results fail more generally.


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Additional Information

Jonathan M. Borwein
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: jborwein@cecm.sfu.ca

James V. Burke
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
Email: burke@math.washington.edu

Adrian S. Lewis
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: aslewis@cecm.sfu.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07149-1
PII: S 0002-9939(03)07149-1
Keywords: Monotonicity, directionally Lipschitz functions, null sets, a.e. differentiability, cones
Received by editor(s): April 11, 2002
Received by editor(s) in revised form: November 19, 2002
Published electronically: July 14, 2003
Additional Notes: The first author’s research was supported by NSERC and by the Canada Research Chair Programme. The second author’s research was supported by NSF DMS-9971852 & NIH P41-RR-12609. The third author’s research was supported by NSERC
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society