Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T13:34:58.916Z Has data issue: false hasContentIssue false
  • English
  • Français

Generic continuity of minimal set-valued mappings

Part of: Normed linear spaces and Banach spaces; Banach lattices Calculus on manifolds; nonlinear operators

Published online by Cambridge University Press:  09 April 2009

W. B. Moors  and
J. R. Giles
W. B. Moors
Affiliation:
Department of Mathematics The University of Auckland, New Zealand
J. R. Giles
Affiliation:
Department of Mathematics The University of NewcastleNSW 2308, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study classes of Banach spaces where every set-valued mapping from a complete metric space into subsets of the Banach space which satisfies certain minimal properties, is single-valued and norm upper semi-continuous at the points of a dense Gδ subset of its domain. Characterisations of these classes are developed and permanence properties are established. Sufficiency conditions for membership of these classes are defined in terms of fragmentability and σ-fragmentability of the weak topology. A characterisation of non membership is used to show that l∞ (N) is not a member of our classe of generic continuity spaces.

MSC classification

Secondary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties 46B20: Geometry and structure of normed linear spaces 58C20: Differentiation theory (Gateaux, Fréchet, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 63 , Issue 2 , October 1997 , pp. 238 - 262
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Borwein, J., Fitzpatrick, S. and Kenderov, P., ‘Minimal convex uscos and monotone operators on small sets’, Canad. J. Math. 43 (1991), 461–476.CrossRefGoogle Scholar
[2]Collier, J. B., ‘The dual of a space with the Radon-Nikodým property’, Pacific J. Math. 64 (1976), 103–106.CrossRefGoogle Scholar
[3]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces, Pitman Monographs Surveys Pure Appl. Math. 64 (Longmans, Harlow, 1993).Google Scholar
[4]Giles, J. R. and Bartlett, M. O., ‘Modified continuity and a generalisation of Michael's Selection Theorem’, Set-Val. Anal. 1 (1993), 365–378.CrossRefGoogle Scholar
[5]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 differentiability theory’, J. Math. Anal. Appl. 178 (1993), 247–268.CrossRefGoogle Scholar
[6]Giles, J. R., ‘The implications for differentiability of a weak index of non-compactness’, Bull. Austral. Math. Soc. 48 (1993), 75–91.CrossRefGoogle Scholar
[7]Giles, J. R., ‘Differentiability properties of Banach spaces where the boundary of the closed unit ball has denting point properties’, Proc. Centre Math. Anal. Austral. Nat. Univ. 29 (1992), 107–115.Google Scholar
[8]Giles, J. R., Kenderov, P. S., Moors, W. B. and Sciffer, S. D., ‘Generic differentiability of convex functions on the dual of a Banach space’, Pacific J. Math. 172 (1996), 413–431.CrossRefGoogle Scholar
[9]Jayne, J. E., Namioka, I. and Rogers, C. A., ‘Topological properties of Banach spaces’, Proc. London Math. Soc. 66 (1993), 651–672.CrossRefGoogle Scholar
[10]Jayne, J. E., ‘σ-fragmentable Banach spaces’, Mathematika 39 (1992), 161–188.CrossRefGoogle Scholar
[11]Jayne, J. E., ‘σ-fragmentable Banach spaces’, Mathematika 39 (1992), 197–215.CrossRefGoogle Scholar
[12]Jayne, J. E., ‘Fragmentability and σ-fragmentabilty’, Fund. Math. 143 (1993), 207–220.CrossRefGoogle Scholar
[13]Kenderov, P. S. and Giles, J. R., ‘On the structure of Banach spaces with Mazur's intersection property’, Math. Ann. 291 (1991), 463–471.CrossRefGoogle Scholar
[14]Moors, W. B., ‘A continuity property related to an index of non-separability and its applications’, Bull. Austral. Math. Soc. 46 (1992), 67–79.CrossRefGoogle Scholar
[15]Namioka, I., ‘Separate continuity and joint continuity’, Pacific J. Math. 51 (1974), 515–531.CrossRefGoogle Scholar
[16]Phelps, R. R., Convex functions, monotone operators and differentiability, Lecture Notes in Math. 1364, 2nd edition, (Springer, Berlin, 1993).Google Scholar
[17]Preiss, David, Phelps, R. R. and Namioka, I., ‘Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings’, Israel J. Math. 72 (1990), 257–279.CrossRefGoogle Scholar
[18]Rosenthall, H., ‘On relatively disjoint families of measures, with some applications to Banach space theory’, Studia Math. 37 (1970), 13–36.CrossRefGoogle Scholar
[19]Smith, M. A., ‘A curious generalisation of local uniform rotundity’, Comment. Math. Univ. Carolin. 25 (1984), 659–665.Google Scholar
[20]Troyanski, S., ‘On a property of the norm which is close to local uniform rotundity’, Math. Ann. 271 (1985), 305–314.CrossRefGoogle Scholar
[21]Troyanski, S., ‘On some generalisations of denting points’, Israel J. Math. 88 (1994), 175–188.CrossRefGoogle Scholar