Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-19T12:12:45.085Z Has data issue: false hasContentIssue false
  • English
  • Français

Locally uniformly rotund norms

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

Published online by Cambridge University Press:  26 February 2010

M. Raja
M. Raja
Affiliation:
Mathçmatiques Pures, Universitç Bordeaux 1351, cours de la Liberation, 33405 Talance Cedex, France. E-mail: matias@um.es
Get access

Abstract

Given a Banach space X and a norming subspace Z⊂X*, a geometrical method is introduced to characterize the existence of an equivalent σ(X, Z)-lsc LUR norm on X. A new simple proof of the Theorem of Troyanski: every rotund space with a Kadec norm is LUR renormable, and a generalization of the Moltó, Orihuela and Troyanski characterization of the LUR renormability, are provided without probability arguments. Among other applications, it is shown that a dual Banach space with a w*-Kadec norm admits a dual LUR norm.

MSC classification

Secondary: 46B07: Local theory of Banach spaces
Type
Research Article
Information
Mathematika , Volume 46 , Issue 2 , December 1999 , pp. 343 - 358
Copyright
Copyright © University College London 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Alexandrov, G. A.. One generalization of the Kadec property. Ganita1 50 (1999), 37–47.Google Scholar
2. Arkangel'skii, A.. An addition theorem for weight of subsets of a compact space (Russian). DokladyAcad. Nauk USSR, 126 (1959), 239–241.Google Scholar
3. Bourgin, R. D.. Geometric aspects of convex sets with the Radon-Nikodym property. Lecture Notes in Math. 993, (Springer-Verlag, 1980).Google Scholar
4. Bessaga, C. and Pelczynski, A.. Selected topics in infinite-dimensional topology. Mono. Mat. 58 (PWN-Polish Scientific Publishers, Warsaw, 1975).Google Scholar
5. Choquet, G.. Lectures on Analysis, Volume II. Math. Lecture Notes 25 (W. A. Benjamin, 1969).Google Scholar
6. Diestel, J.. Geometry of Banach spaces: Selected topics. Lecture Notes in Math. 485 (Springer-Verlag, 1975).CrossRefGoogle Scholar
7. Deville, R., Godefroy, G. and Zizler, V.. Smoothness and Renorming in Banach Spaces. (Pitman Monographs and Surveys 64, 1993).Google Scholar
8. Fabian, M.. On a dual locally uniformly rotund norm on a dual VaŠak space. Studia Math. 101 (1991), 69–81.CrossRefGoogle Scholar
9. Fabian, M. and Godefroy, G.. The dual of every Asplund space admits a projectional resolution of the identity. Studia Math., 91 (1988), 141–151.CrossRefGoogle Scholar
10. Haydon, R.. Countable unions of compact spaces with the Namioka property. Mathematika, 41 (1994), 141–144.CrossRefGoogle Scholar
11. Haydon, R.. Bad trees, bad norms and the Namioka property. Mathematika, 42 (1995), 30–42.CrossRefGoogle Scholar
12. Haydon, R.. Trees in renorming theory. Proc. London Math. Soc, 78 (1999), 541–584.CrossRefGoogle Scholar
13. Jayne, J. E., Namioka, I. and Rogers, C. A., σ-fragmentable Banach spaces. Mathematika, 39 (1992), 161–188 and 197-215.CrossRefGoogle Scholar
14. Lancien, G.. Thçorie de l'indice et problemes de renormage en géométrie des espaces de Banach. (Thése, Paris, 1992).Google Scholar
15. Lin, B. L., Lin, P. K. and Troyanski, S. L.. Characterizations of denting points. Proc. Amer. Math. Soc, 102 (1988), 526–528.CrossRefGoogle Scholar
16. Molto, A., Orihuela, J. and Troyanski, S.. Locally uniform rotund renorming and fragmentability. Proc. London Math. Soc, 75 (1997), 619–640.CrossRefGoogle Scholar
17. Namioka, I. and Phelps, R. R.. Banach spaces which are Asplund spaces. Duke Math. J., 42 (1975), 735–750.CrossRefGoogle Scholar
18. Raja, M.. Kadec norms and Borel sets in a Banach space. Studia Math., 136 (1999), 1–16.CrossRefGoogle Scholar
19. Raja, M.. On topology and renorming of a Banach space. C. R. Acad. Bulgare Sci., 136 (1999), 13–16.Google Scholar
20. Vaİak, L.. On one generalization of weakly compact generated spaces. Studia Math., 70 (1981), 11–19.Google Scholar