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Normes infiniment différentiables sur certains espaces de Banach. (Infinitely differentiable norms on certain Banach spaces). (French) Zbl 0788.46008
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)].

##### MSC:
 46B03 Isomorphic theory (including renorming) of Banach spaces
##### Keywords:
differentiable norm; Talagrand’s construction