$C^k$-smooth approximations of LUR norms

Petr Hajek (1) , Antonin . Prochazka (2, 3)

1 IM / CAS - Institute of Mathematics of the Czech Academy of Science
2 IMB - Institut de Mathématiques de Bordeaux
3 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

Let $X$ be a WCG Banach space admitting a $C^k$-Fr\' echet smooth norm. Then $X$ admits an equivalent norm which is simultaneously $C^1$-Fr\' echet smooth, LUR, and a uniform limit of $C^k$-Fr\' echet smooth norms. If $X=C([0,\alpha])$, where $\alpha$ is an ordinal, then the same conclusion holds true with $k=\infty$.

Mots clés: LUR; higher order differentiability; renorming

Domaines: Analyse fonctionnelle [math.FA]

Petr Hajek, Antonin . Prochazka. $C^k$-smooth approximations of LUR norms. Transactions of the American Mathematical Society, 2014, 366, pp.1973-1992. ⟨10.1090/S0002-9947-2013-05899-0⟩. ⟨hal-00355409⟩