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Locally Uniformly Rotund Renorming and Injections Into c0(Γ)

Published online by Cambridge University Press:  20 November 2018

G. Godefroy ,
S. Troyanski ,
J. Whitfield  and
V. Zizler
G. Godefroy
Affiliation:
Équipe D'analyse, Université Paris VI, Place Jussieu, F-75230, Paris, Cedex 05, France
S. Troyanski
Affiliation:
Department of Mathematics, University of Sofia, Boul. A. Ivanov 5, Sofia, Bulgaria
J. Whitfield
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada P7B5E1
V. Zizler
Affiliation:
Institut für Angewandte Mathematik der Universität, Wegelerstr. 6, D-5300 Bonn, West Germany Department of Mathematics, University of Alberta, Edmonton T6G261 Alberta, Canada
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Abstract

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A norm |⋅| on a Banach space X is locally uniformly rotund (LUR) if lim |xn — x| = 0 for every xn, x ∈ X for which lim2|x|2 + 2 |xn|2-|xn+xn|2 = 0. It is shown that a Banach space X admits an equivalent LUR norm provided there is a bounded linear operator T of X into c0(Γ) such that T* c*0(Γ) is norm dense in X*. This is the case e.g. if X* is weakly compactly generated (WCG).

Keywords

46B 20renorminglocal uniform rotunditydifferentiability of the norm
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 4 , 01 December 1984 , pp. 494 - 500
Copyright
Copyright © Canadian Mathematical Society 1984

Footnotes

*

This author's research supported in part by NSERC (Canada) Grant A7535.

*

This author's research supported by Sonderforschungsbereich 72 der Universität Bonn (West Germany).

References

1. Day, M. M., Normed linear spaces, 3rd. ed. Springer-Verlag, New York, 1973,10.1007/978-3-662-09000-8CrossRefGoogle Scholar
2. Diestel, J., Geometry of Banach spaces. Selected topics, Lecture Notes in Math. Vol. 485, Springer-Verlag, New York, 1975,Google Scholar
3. Godefroy, G., Troyanski, S., Whitfield, J. and Zizler, V., Smoothness in weakly compactly generated Banach spaces, J. Funct. Anal. 52 (1983), 344-352.Google Scholar
4. John, K. and Zizler, V., Markusevic bases in some dual spaces, Proc. Amer. Math. Soc. 50 (1975), 293-296,Google Scholar
5. Rainwater, J., Local uniform convexity of Day's norm on c0(T), Proc. Amer. Math. Soc. 22 (1969), 335-339,Google Scholar
6. Rosenthal, H. P., The heredity problem for weakly compactly generated Banach spaces, Compositio Math. 28 (1974), 83-111,Google Scholar
7. Troyanski, S., On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173-180.CrossRefGoogle Scholar