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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

$ C^k$-smooth approximations of LUR norms


Authors: Petr Hájek and Antonín Procházka
Journal: Trans. Amer. Math. Soc. 366 (2014), 1973-1992
MSC (2010): Primary 46B20, 46B03, 46E15
Published electronically: December 13, 2013
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Abstract: Let $ X$ be a WCG Banach space admitting a $ C^{k}$-smooth norm where $ k \in \mathbb{N} \cup \left \{\infty \right \}$. Then $ X$ admits an equivalent norm which is simultaneously, $ C^1$-smooth, LUR, and the limit of a sequence of $ C^{k}$-smooth norms. If $ X=C([0,\alpha ])$, where $ \alpha $ is any ordinal, then the same conclusion holds true with $ k=\infty $.


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Additional Information

Petr Hájek
Affiliation: Mathematical Institute, Czech Academy of Science, Žitná 25, 115 67 Praha 1, Czech Republic – and – Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 166 27 Prague 6, Czech Republic
Email: hajek@math.cas.cz

Antonín Procházka
Affiliation: Laboratoire de Mathématiques UMR 6623, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon Cedex, France
Email: antonin.prochazka@univ-fcomte.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05899-0
PII: S 0002-9947(2013)05899-0
Keywords: LUR, smoothness, higher order smoothness, renorming
Received by editor(s): January 22, 2009
Received by editor(s) in revised form: April 4, 2011, May 3, 2012, and June 19, 2012
Published electronically: December 13, 2013
Additional Notes: This work was supported by grants GA CR Grant P201/11/0345, RVO: 67985840, and PHC Barrande 2012 26516YG
Article copyright: © Copyright 2013 by the authors