Some remarks on smooth renormings of Banach spaces

We prove that in every separable Banach space X with a Schauder basis and a Ck-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a Ck-smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail o...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications Vol. 455; no. 2; pp. 1272 - 1284
Main Authors: Hájek, Petr, Russo, Tommaso
Format: Journal Article
Language:English
Published: Elsevier Inc 15.11.2017
Subjects:
[formula omitted]-smooth norm
Approximation of norms
Fréchet smooth
Implicit function theorem
Minkowski functional
Renorming
Renorming
Fréchet smooth
Implicit function theorem
Ck-smooth norm
Approximation of norms
Minkowski functional
ISSN:0022-247X, 1096-0813
Online Access:Get full text
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Summary:We prove that in every separable Banach space X with a Schauder basis and a Ck-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a Ck-smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the Schauder basis. Our result solves a problem from the recent monograph of Guirao, Montesinos and Zizler.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2017.05.077