Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T21:47:08.326Z Has data issue: false hasContentIssue false
  • English
  • Français

On uniformly Gâteaux differentiable norms in C(K)

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

Published online by Cambridge University Press:  26 February 2010

A. Moltó  and
S. Troyanski
A. Moltó
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, Dr. Moliner 50, 46100-Burjasot (Valencia), Spain.
S. Troyanski
Affiliation:
Department of Mathematics and Informatics, Sofia University, 5, James Bourchier Blvd., 1126 Sofia, Bulgaria.
Get access

Abstract

It is proved that C(K) has no equivalent uniformly Gâteaux differentiable norm (UGD) when K-is an uncountable separable scattered com¬pact space. This result is applied to obtain an example of scattered compact K such that K′″ = ∅ and C(K) has no UGD renorming.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Mathematika , Volume 41 , Issue 2 , December 1994 , pp. 233 - 238
Copyright
Copyright © University College London 1994

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

D.Deville, R.. Problèmes de renormages. J. Functional Anal., 68 (1986), 117–129.CrossRefGoogle Scholar
DGZ.Deville, R., Godefroy, G. and Zizler, V.. Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Appl. Math., 64 (Longman Scientific & Technical, Harlow, 1993).Google Scholar
GJ.Gillman, L. and Jerison, M.. Rings of Continuous Functions (Van Nostrand Reinhold Company, 1960).CrossRefGoogle Scholar
H.Haydon, R.. Trees in renorming theory. Preprint, April, 1992.Google Scholar
HR.Haydon, R. G. and Rogers, C. A.. A locally uniformly convex renorming for certain C(K). Mathematika, 37 (1990), 1–8.CrossRefGoogle Scholar
M.Molto, A.. On a theorem of Sobczyk. Bull. Australian Math. Soc., 43 (1991), 123–130.CrossRefGoogle Scholar
T.Troyanski, S.. Construction of equivalent norms for certain local characteristics with rotundity and smoothness by means of martingales. Proceedings of the 14th Spring Conference of the Union of Bulgarian Mathematics, 1985, 129–156. (Russian).Google Scholar