Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T16:06:39.470Z Has data issue: false hasContentIssue false
  • English
  • Français

On Locally Uniformly Rotund Renormings in C(K) Spaces

Published online by Cambridge University Press:  20 November 2018

J. F. Martínez ,
A. Moltó ,
J. Orihuela  and
S. Troyanski
J. F. Martínez
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Valencia, Valencia, Spain, e-mail: j.francisco.martinez@uv.es, anibal.molto@uv.es
A. Moltó
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Murcia, Spain, e-mail: joseori@um.es, stroya@um.es
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.

A characterization of the Banach spaces of type $C\left( K \right)$ that admit an equivalent locally uniformly rotund norm is obtained, and a method to apply it to concrete spaces is developed. As an application the existence of such renorming is deduced when $K$ is a Namioka–Phelps compact or for some particular class of Rosenthal compacta, results which were originally proved with ad hoc methods.

Keywords

46B0346B20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 3 , 01 June 2010 , pp. 595 - 613
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Deville, R., Godefroy, G., and Zizler, V., Smoothness and Renormings in Banach Spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics 64. Longman Scientific & Technical, Harlow, 1993.Google Scholar
[2] Garćıa, F., Oncina, L., Orihuela, J., and Troyanski, S., Kuratowski's index of non-compactness and renorming in Banach spaces. J. Convex Anal. 11(2004), no. 2, 477–494.Google Scholar
[3] Haydon, R., Locally uniformly rotund norms in Banach spaces and their duals, J. Funct. Anal. 254(2008), no. 8, 2023–2039.Google Scholar
[4] Haydon, R., Trees in renorming theory. Proc. London Math. Soc. 78(1999), no. 3, 541–584. doi:10.1112/S0024611599001768 Google Scholar
[5] Haydon, R., Jayne, J. E., Namioka, I., and Rogers, C. A., Continuous functions on totally ordered spaces that are compact in their order topologies. J. Funct. Anal. 178(2000), no. 1, 23–63. doi:10.1006/jfan.2000.3652 Google Scholar
[6] Haydon, R., Moltó, A., and Orihuela, J., Spaces of functions with countably many discontinuities. Israel J. Math. 158(2007), 19–39. doi:10.1007/s11856-007-0002-1 Google Scholar
[7] Martınez, J. F., Renormings in C(K) Spaces, Universidad de Valencia, Ph.D. dissertation, 2007.Google Scholar
[8] Moltó, A., Orihuela, J., Troyanski, S., and Valdivia, M., A Non Linear Transfer Technique for Renorming. Lectures Notes in Mathematics 1951. Springer-Verlag, Berlin, 2009.Google Scholar
[9] Phelps, R. R., Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics 1364. Springer–Verlag, Berlin, 1989.Google Scholar
[10] Raja, M., Locally uniformly rotund norms. Mathematika 46(1999), no. 2, 343–358.Google Scholar
[11] Raja, M., On dual locally uniformly rotund norms. Israel J. Math. 129(2002), 77–91. doi:10.1007/BF02773154 Google Scholar
[12] Todorčević, S., Topics in Topology. Lecture Notes in Mathematics 1652. Springer-Verlag, Berlin, 1997.Google Scholar