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Convex functions on Banach spaces not containing ℓ1

  • Jon Borwein (a1) and Jon Vanderwerff (a2)
Abstract

There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an isomorphic copy of ℓ1. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions.

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References
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1. Beer, G., Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, The Netherlands, 1993.
2. Beer, G. and Borwein, J., Mosco and slice convergence of level sets and graphs of linear functionals, J. Math. Anal. Appl. 175 (1993), 5367.
3. Borwein, J., Asplund spaces are “sequentially reflexive”, CORR 91–14, University of Waterloo, 1991.
4. Borwein, J. and Fabian, M., On convex functions having points of Gateaux differentiability which are not points of Fréchet differentiability, Canad. J. Math. 45 (1993), 11211134.
5. Borwein, J., Fabian, M. and Vanderwerff, J., Locally Lipschitz functions and bornological derivatives, CECM Research Report 93–012, Simon Fraser University, 1993.
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7. Borwein, J. and Vanderwerff, J., Epigraphical and uniform convergence of convex functions, Trans. Amer. Math. Soc. 348 (1996), 16171631.
8. Diestel, J., Sequences and Series in Banach Spaces, Graduate Texts in Mathematics 92, Springer-Verlag, Berlin–New York–Tokyo, 1984.
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10. Ørno, P., On J. Borwein's concept of sequentially reflexive Banach spaces, Banach Bulletin Board, 1991.
11. Phelps, R. R., Convex Functions, Monotone Operators and Differentiability, Lecture Notes inMathematics 1364, Springer–Verlag, 1989.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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