## $C^k$-smooth approximations of LUR norms

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- by Petr Hájek and Antonín Procházka PDF
- Trans. Amer. Math. Soc.
**366**(2014), 1973-1992

## 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 $k=\infty$.## References

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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
- 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
- © Copyright 2013 by the authors
- Journal: Trans. Amer. Math. Soc.
**366**(2014), 1973-1992 - MSC (2010): Primary 46B20, 46B03, 46E15
- DOI: https://doi.org/10.1090/S0002-9947-2013-05899-0
- MathSciNet review: 3152719