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On Convex Functions Having Points of Gateaux Differentiability Which are Not Points of Fréchet Differentiability

Published online by Cambridge University Press:  20 November 2018

J. M. Borwein  and
M. Fabian
J. M. Borwein*
Affiliation:
Department of Combinatorics and Optimization University of Waterloo, Waterloo, Ontario N2L 3G1
M. Fabian*
Affiliation:
Department of Mathematics Faculty of Machine Engineering Czech Technical University12800 Prague 2 Czechoslovakia
*
Current address: Department of Mathematics and Statistics Simon Fraser University Burnaby, British Columbia V5A 1S6
Current address: Department of Mathematics and Statistics Simon Fraser University Burnaby, British Columbia V5A 1S6
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Abstract

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We study the relationships between Gateaux, Fréchet and weak Hadamard differentiability of convex functions and of equivalent norms. As a consequence we provide related characterizations of infinite dimensional Banach spaces and of Banach spaces containing ł1. Explicit examples are given. Some renormings of WCG Asplund spaces are made in this vein.

Keywords

46G0546B0349J5058C2046A1746B2052A41Key words and phrases:Gateaux differentiabilityFréchet differentiabilityweak Hadamard differentiability(not)containing l1renormingweak*Kadec propertyconvex functionsnon-compact operators
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 6 , 01 December 1993 , pp. 1121 - 1134
Copyright
Copyright © Canadian Mathematical Society 1993

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