$C^k$-smooth approximations of LUR norms

Abstract

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 $k=\infty$.