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A CONTINUITY CHARACTERIZATION OF ASPLUND SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  07 February 2011

J. R. GILES
J. R. GILES*
Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, New South Wales 2308, Australia (email: [email protected])
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Abstract

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A Banach space is an Asplund space if every continuous gauge has a point where the subdifferential mapping is Hausdorff weak upper semi-continuous with weakly compact image. This contributes towards the solution of a problem posed by Godefroy, Montesinos and Zizler.

Keywords

Fréchet differentiabilitysubdifferential mappingweak upper semi-continuitygaugeextreme pointpolarRadon–Nikodým property

MSC classification

Secondary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 450 - 455
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Benítez, J. and Montesinos, V., ‘Restricted weak upper semicontinuous differentials of convex functions’, Bull. Aust. Math. Soc. 63 (2001), 93–100.CrossRefGoogle Scholar
[2]Bourgin, R. D., Geometric Aspects of Convex Sets with the Radon–Nikodým Property, Springer Lecture Notes in Mathematics, 33 (Springer, New York, 1983).CrossRefGoogle Scholar
[3]Contreras, M. D. and Payá, R., ‘On upper semicontinuity of duality mappings’, Proc. Amer. Math. Soc. 121 (1994), 451–459.CrossRefGoogle Scholar
[4]Fabian, M., Habala, P., Hájek, P., Santalucía, V. M., Pelant, J. and Zizler, V., Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics (Springer, New York, 2001).CrossRefGoogle Scholar
[5]Franchetti, C. and Payá, R., ‘Banach spaces with strongly subdifferentiable norm’, Boll. Unione Mat. Ital. 7 (1993), 45–70.Google Scholar
[6]Giles, J. R., Gregory, D. A. and Sims, B., ‘Geometrical implications of upper semi-continuity of the duality mappings on Banach space’, Pacific J. Math. 79 (1978), 99–109.CrossRefGoogle Scholar
[7]Giles, J. R. and Moors, W. B., ‘A continuity property related to Kuratowski’s index of non-compactness, its relevance to the drop property, and its implications for differentiabilty theory’, J. Math. Anal. Appl. 178 (1993), 247–268.CrossRefGoogle Scholar
[8]Giles, J. R. and Moors, W. B., ‘Generic continuity of restricted weak upper semi-continuous set-valued mappings’, Set-Valued Anal. 4 (1996), 25–39.CrossRefGoogle Scholar
[9]Godefroy, G., ‘Some applications of Simons’ inequality’, Serdica. Math. J. 26 (2000), 59–78.Google Scholar
[10]Godefroy, G. and Indumathi, V., ‘Norm-to-weak upper semi-continuity of the duality and pre-duality mappings’, Set-Valued Anal. 10 (2002), 317–330.CrossRefGoogle Scholar
[11]Godefroy, G., Indumathi, V. and Lust-Piquard, F., ‘Strong subdifferentiability of convex functionals and proximinality’, J. Approx. Theory 116 (2002), 397–415.CrossRefGoogle Scholar
[12]Godefroy, G., Montesinos, V. and Zizler, V., ‘Strong subdifferentiability of norms and geometry of Banach spaces’, Comment Math. Uni. Carolin. 36 (1995), 493–502.Google Scholar
[13]Phelps, R. R., Convex Functions, Monotone Operators and Differentiability, 2nd edn, Lecture Notes in Mathematics, 1364 (Springer, New York, 1993).Google Scholar