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Journal of the London Mathematical Society Advance Access originally published online on July 11, 2007
Journal of the London Mathematical Society 2007 75(3):647-658; doi:10.1112/jlms/jdm040
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© 2007 London Mathematical Society

Strictly convex renormings

A. Moltó1,, J. Orihuela2, S. Troyanski3 and V. Zizler4

1 Departamento de Análisis Matemático
Facultad de Matemáticas
Universidad de Valencia
Dr. Moliner 50
46100 Burjasot (Valencia)
Spain
2 Departamento de Matemáticas
Universidad de Murcia
Campus de Espinardo
30100 Espinardo
Murcia
Spain
joseori{at}um.es
3 Departamento de Matemáticas
Universidad de Murcia
Campus de Espinardo
30100 Espinardo
Murcia
Spain
stroya{at}um.es
4 Mathematical Institute
Czech Academy of Sciences
Zitna 25
11567 Praha 1
Prague
Czech Republic
zizler{at}math.cas.cz

A normed space X is said to be strictly convex if x = y whenever ||(x + y)/2|| = ||x|| = ||y, in other words, when the unit sphere of X does not contain non-trivial segments. Our aim in this paper is the study of those normed spaces which admit an equivalent strictly convex norm. We present a characterization in linear topological terms of the normed spaces which are strictly convex renormable. We consider the class of all solid Banach lattices made up of bounded real functions on some set {Gamma}. This class contains the Mercourakis space c1({Sigma}' x {Gamma}) and all duals of Banach spaces with unconditional uncountable bases. We characterize the elements of this class which admit a pointwise strictly convex renorming.

anibal.molto{at}uv.es

2000 Mathematics Subject Classification 35J25 (primary), 28C15 (secondary).

The first author has been supported by BFM2003–07540/MATE, Ministerio de Ciencia y Tecnología MCIT y FEDER (Spain). The second author is supported by DGES, Project BFM2002–01719 (Spain) and Fundación Séneca 00690/PI/04 CARM (Spain). The third author is supported by Project of Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and grant MM-1401/04 of Bulgarian NFR; by DGES, Project BFM2002–01719 (Spain) and Fundación Séneca 00690/PI/04 CARM (Spain). The fourth author is supported by the AS CR Institutional Research Plan No. AV0Z10190503 and by the research grant A 100190502 (Czech Republic).

Received August 22, 2005; revised September 27, 2006; published online July 11, 2007.


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