Haydon, Richard Normes infiniment différentiables sur certains espaces de Banach. (Infinitely differentiable norms on certain Banach spaces). (French) Zbl 0788.46008 C. R. Acad. Sci., Paris, Sér. I 315, No. 11, 1175-1178 (1992). We say that the norm in the Banach space \(B\) is class \(C^ k\) if the function \(x\to \| x\|\) is class of \(C^ k\) except 0. Let \(L\) be a locally compact homeomorphic to a finite product of intervals of ordinals, for instance a countable locally compact space.The author of this note constructs a norm on \(C_ 0(L)\), equivalent to the supremum norm, which is infinitely differentiable except 0. He also shows that if a Banach space \(E\) has a norm that is \(k\) times continuously differentiable, then the space \(C_ 0(L;E)\) has an equivalent norm with the same order of smoothness. The results are based on some construction which we can trace back to M. Talagrand’s construction [Isr. J. Math., 54, 327-334 (1986; Zbl 0611.46023)]. Reviewer: S.Koshi (Utsunomiya) Cited in 3 ReviewsCited in 8 Documents MSC: 46B03 Isomorphic theory (including renorming) of Banach spaces Keywords:differentiable norm; Talagrand’s construction Citations:Zbl 0611.46023 PDFBibTeX XMLCite \textit{R. Haydon}, C. R. Acad. Sci., Paris, Sér. I 315, No. 11, 1175--1178 (1992; Zbl 0788.46008)