𝐶^{𝑘}-smooth approximations of LUR norms

Hájek, Petr; Procházka, Antonín

doi:10.1090/S0002-9947-2013-05899-0

$C^k$-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 $k=\infty$.

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