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Subdifferentials Whose Graphs Are Not Norm × Weak* Closed

Published online by Cambridge University Press:  20 November 2018

Jonathan Borwein ,
Simon Fitzpatrick  and
Roland Girgensohn
Jonathan Borwein
Affiliation:
Department of Mathematics Simon Fraser University Burnaby, British Columbia V5A 1S6, e-mail: jborwein@cecm.sfu.ca
Simon Fitzpatrick
Affiliation:
Department of Mathematics University of Western Australia Nedlands, Western Australia 6009 Australia, e-mail: fitzpatr@maths.uwa.edu.au
Roland Girgensohn
Affiliation:
GSF-Forschungszentrum Postfach 1129 85758 Neuherberg Germany, e-mail: girgen@gsf.de
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Abstract

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In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a proper lower semicontinuous convex function on a separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed.

Keywords

46N1047H05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 4 , 01 December 2003 , pp. 538 - 545
Copyright
Copyright © Canadian Mathematical Society 2003

References

[B-F] Borwein, J. M. and Fitzpatrick, S., Local boundedness of monotone operators under minimal hypotheses. Bull. Austral.Math. Soc. 39 (1989), 439–441.Google Scholar
[B-F2] Borwein, J. M. and Fitzpatrick, S., w*-sequential compactness and bornological limit derivatives. J. Convex Anal.: Special Issue in Celebration of R. T. Rockafellar's 60th Birthday. Part I, 2 (1995), 59–68.Google Scholar
[B-F-K] Borwein, J. M., Fitzpatrick, S. and Kenderov, P.,Minimal convex usco mappings and monotone operators on small sets. Canad. J. Math. 43 (1991), 461–476.Google Scholar
[B-L] Borwein, J. M. and Lewis, A. S., Convex Analysis and Nonlinear Optimization. CMS Books Math. 3, Springer-Verlag, New York, 2000.Google Scholar
[Cl] Clarke, F. H., Optimization and Nonsmooth Analysis. SIAM, 1990, originally published by John Wiley, 1983.Google Scholar
[F-H-H] Fabian, M., Habala, P., Hájek, P., Santalucia, V. Montesinos, Pelant, J. and Zizler, V., Functional Analysis and Infinite-Dimensional Geometry. CMS Books Math. 8, Springer-Verlag, New York, 2001.Google Scholar
[Ho] Holmes, R. B., Geometric Functional Analysis and its Applications. Graduate Texts in Math. 24, Springer-Verlag, New York, 1975.Google Scholar
[Ko] Kolomý, J., Maximal monotone and accretive multivalued mappings and the structure of Banach spaces. 170–177, In: Function spaces, Poznan 1986, Teubner, Leipzig, 1988.Google Scholar
[Mo] Mordukhovich, B. S., The extremal principle and its applications to optimization and economics. In: Optimization and Related Topics, (eds., A. Rubinov and B. Glover), Appl. Optim. 37, Kluwer, 343–369.Google Scholar
[M-S] Mordukhovich, B. S. and Shao, Y., Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348 (1996), 1235–1280.Google Scholar
[Ph] Phelps, R. R., Convex functions, Monotone Operators and Differentiability. 2nd ed., Springer-Verlag, Berlin, 1993.Google Scholar
[R-W] Rockafellar, R. T. and Wets, R. J.-B., Variational Analysis. Springer-Verlag, New York, 1998.Google Scholar
[Si] Simons, S., Minimax and Monotonicity. Lecture Notes in Math. 1693, Springer-Verlag, Berlin, 1998.Google Scholar