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On a Banach space without a weak mid-point locally uniformly rotund norm

Published online by Cambridge University Press:  17 April 2009

Zhibao Hu ,
Warren B. Moors  and
Mark A. Smith
Zhibao Hu
Affiliation:
Division of Mathematics, Science and Exss, El Paso Community College, Valle Verde Campus, PO Box 20500 El Paso, Texas 79998, United States of America
Warren B. Moors
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019 Auckland, New Zealand
Mark A. Smith
Affiliation:
Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056, United States of America
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Abstract

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In this paper we show that (i) l∞ does not admit an equivalent weak mid-point locally uniformly rotund norm and (ii) l∞ / c0 does not admit an equivalent rotund norm.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 2 , October 1997 , pp. 193 - 196
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Aleksandrov, G.A. and Dimitrov, I.P., ‘On the equivalent weakly midpoint locally uniformly rotund renorming of the space l∞’, (in Russian), in Mathematics and Mathematical Education (Sunny Beach, 1985), pp. 189–191.Google Scholar
[2]Bourgain, J., ‘l ∞ / c 0 has no equivalent strictly convex norm’, Proc. Amer. Math. Soc. 78 (1980), 225–226.Google Scholar
[3]Lindenstrauss, J., ‘Weakly compact sets – their topological properties and the Banach spaces they generate’, Ann. of Math. Studies 69 (1972), 235–273.Google Scholar
[4]Smith, M.A., ‘A curious generalisation of local uniform rotundity’, Comment Math. Univ. Carol 25 (1984), 659–665.Google Scholar