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

الأصل · EN

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 Cᵐ and of bounded support, then all continuous real-valued functions on C(K) may be uniformly approximated by functions of class Cᵐ. (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 Cⁱnfty (except at 0).