2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217213Let $X$ 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 $X=C([0,α])$, where $α$ is an ordinal, then the same conclusion holds true with $k=\infty$.Functional Analysis46B20, 46B03, 46E15text