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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

References

[B] Borwein, J.M., Asplund spaces are ‘Sequentially reflexive ‘ , preprint.Google Scholar
[BFa] Borwein, J.M. and Fabian, M., On convex functions having points of Gateaux differentiability which are not points of Fréchet differentiability , Technical Report, University of Waterloo CORR 92-04, February 1992.Google Scholar
[BF] Borwein, J.M. and Fitzpatrick, S., A weak Hadamard smooth renorming of L1 (Ω, μ) , Canad. Math. Bull., to appear.Google Scholar
[Dl] Diestel, J., Geometry of Banach spaces—Selected topics, Lect. Notes in Mathematics 485, Springer- Verlag, 1975.Google Scholar
[D2] Diestel, J., Sequences and series in Banach spaces, Graduate texts in Mathematics, Springer Verlag, N.Y., Berlin, Tokyo, 1984.Google Scholar
[DGZ] Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings and differentiability in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, J. Wiley & Sons, Inc., New York 1993.Google Scholar
[E] Emmanuele, G., A dual characterization of Banach spaces not containing l\, Bull. Polish Acad. Sc, 34(1986), 155–159.Google Scholar
[GMS] Ghoussoub, N.,Maurey, B. and Schachermayer, W., Geometrical implications of certain infinite dimensional decompositions, Trans. Amer. Math. Soc. 317(1990), 541–584.Google Scholar
[H] Haydon, R., A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 20(1990), 261–268.Google Scholar
[K] Klee, V., Two renorming constructions related to a question ofAnselone, Studia Math. 33(1969), 231–242.Google Scholar
[KP] Kadec, M.I. and Pelczyhski, A., Basic sequences, biorthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25(1965), 297–323.(Russian).Google Scholar
[LT] Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces I, Springer Verlag, Berlin, Heidelberg, N.Y., 1977.Google Scholar
[M] Mil, V.D.'man, Geometric theory of Banach spaces, Part I, Russian Math. Surveys 25(1970), 111–170.Google Scholar
[NP] Namioka, J., Phelps, R.R., Banach spaces which are Asplund spaces, Duke Math. 42(1975), 735–750.Google Scholar
[ø] Ørno, P., On J. Borwein's concept of sequentially reflexive Banach spaces , preprint.Google Scholar
[Ph] Phelps, R.R., Convex functions, Monotone operators and differentiability, Lecture Notes in Mathematics 1364, Springer Verlag, 1988.Google Scholar
[T] Talagrand, M., Renormages des quelques C(K), Israel J. Math. 54(1986), 327–334.Google Scholar
[R] Rosenthal, H.P., On quasicomplemented subspaces ofBanach spaces, with an appendix on conjectures of operators from Lp (μ) to Lr(ν), J. Funct. Anal. 4(1969. 176–214.Google Scholar
[V] Vanderwerff, J., Extensions of Markusevic bases, preprint.Google Scholar