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A selection theorem for weak upper semi-continuous set-valued mappings

Published online by Cambridge University Press:  17 April 2009

Warren B. Moors
Warren B. Moors
Affiliation:
Department of MathematicsThe University of AucklandPrivate Bag 92019AucklandNew Zealand
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Abstract

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Let Φ be a set-valued mapping from a Baire space T into non-empty closed subsets of a Banach space X, which is upper semi-continuous with respect to the weak topology on X. In this paper, we give a condition on T which is sufficient to ensure that Φ admits a selection which is norm continuous at each point of a dense and Gδ subset of T. We also derive a variation of James' characterisation of weak compactness, which we use in conjunction with our selection theorem, to deduce some differentiability results for continuous convex functions defined on dual Banach spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 53 , Issue 2 , April 1996 , pp. 213 - 227
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Bourgain, J. and Talagrand, M., ‘Compacité extrémale’, Proc. Amer. Math. Soc. 80 (1980), 68–70.CrossRefGoogle Scholar
[2]Choquet, G., Lectures on analysis 1 (Benjamin, New York and Amsterdam, 1969).Google Scholar
[3]Christensen, J.P.R., ‘Joint continuity of separately continuous functions’, Proc. Amer. Math. Soc. 82 (1981), 455–461.CrossRefGoogle Scholar
[4]Giles, J.R., Gregory, D.A. and Sims, B., ‘Geometrical implications of upper semi-continuity of the duality mapping on a Banach space’, Pacific J. Math. 79 (1978), 99–109.Google Scholar
[5]James, R.C., ‘Reflexivity and the supremum of linear functionals’, Israel J. Math. 13 (1972), 289–230.Google Scholar
[6]Kurana, S.S., ‘Pointwise compactness on extreme points’, Proc. Amer. Math. Soc. 83 (1981), 347–348.CrossRefGoogle Scholar
[7]Namioka, I., ‘Separate continuity and joint continuity’, Pacific J. Math. 51 (1974), 515–531.CrossRefGoogle Scholar
[8]Phelps, R.R., ‘Convex functions, monotone operators and differentiability’, Lecture Notes in Mathematics 1364 (Springer-Verlag, Berlin, Heidelberg, New York, 1993).Google Scholar
[9]Saint-Raymond, J., ‘Jeux topologique et espaces de Namioka’, Proc. Amer. Math. Soc. 87 (1983), 499–504.Google Scholar
[10]Stegall, C., ‘Generalizations of a theorem of Namioka’, Proc. Amer. Math. Soc. 102 (1988), 559–564.Google Scholar
[11]Stegall, C., ‘Functions of the first Baire class with values in Banach spaces’, Proc. Amer. Math. Soc. 111 (1991), 981–991.Google Scholar