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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)].

46B03 Isomorphic theory (including renorming) of Banach spaces