Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-21T13:17:18.706Z Has data issue: false hasContentIssue false
  • English
  • Français

A Dual characterization of Banach Spaces With the Convex Point-of-Continuity Property

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

Published online by Cambridge University Press:  20 November 2018

D. E. G. Hare
D. E. G. Hare*
Affiliation:
Department of Mathematics University of British Columbia Vancouver, Canada
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 introduce a new type of differentiability, called cofinite Fréchet differentiability. We show that the convex point-of-continuity property of Banach spaces is dual to the cofinite Fréchet differentiability of all equivalent norms. A corresponding result for dual spaces with the weak* convex point-of-continuity property is also established.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 3 , 01 September 1989 , pp. 274 - 280
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Asplund, E., Fréchet differentiability of convex functions, Acta Math. 121 (1968), 735–750.Google Scholar
2. Bourgain, J., Dentability and finite-dimensional decompositions, Studia Math. 67 (1980), 135–148.Google Scholar
3. Cudia, D., The geometry of Banach spaces. Smoothness, Trans. Amer. Math. Soc. 110 (1964), 284– 314.Google Scholar
4. Deville, R., G. Godefroy, D. E. G. Hare and V. Zizler, Differentiability of convex functions and the convex point of continuity property in Banach spaces, Israel J. Math., 59 (1957), 245–255.Google Scholar
5. Diestel, J. and Uhl, J. J. Jr., Vector Measures, Math Surveys, No. 15, American Math Society, Providence, R. I., 1977.Google Scholar
6. Ghoussoub, N., Maurey, B. and W. Schachermayer, Geometrical implications of certain infinite dimensional decompositions, to appear.Google Scholar
7. Huff, R. E. and P. D. Morris, Geometric characterizations of the Radon-Nikodym property in Banach spaces, Studia Math. 56 (1976), 157–164.Google Scholar
8. John, K. and V. Zizler, A note on strong differentiability spaces, Comment. Math. Univ. Carolinae 17 (1976), 127–134.Google Scholar
9. Namioka, I. and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), 735–750.Google Scholar
10. Phelps, R. R., Dentability and extreme points in Banach spaces, J. Funct. Anal. 17 (1974), 78–90.Google Scholar
11. Smuljan, V., Sur la dérivabilité de la norme dans l'espace de Banach, Dokl. Akad. Nauk SSSR 27 (1940), 643–648.Google Scholar
12. Stegall, C., The Radon-Nikodym property in conjugate Banach spaces, II, Trans. Amer. Math. Soc. 206 (1975), 213–223.Google Scholar