Smooth norms and approximation in Banach spaces of the type C(K)

Details Smooth norms and approximation in Banach spaces of the type C(K) Smooth norms and approximation in Banach spaces of the type C(K)

2006-10-12

arxiv.org/abs/math/0610421v1

Mathematics

Functional Analysis

Scientific paper

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be uniformly approximated by functions of class C^m. (ii) If C(K) admits an equivalent norm with locally uniformly convex dual norm, then C(K) admits an equivalent norm which is of class C^infty (except at 0).

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