Smooth norms and approximation in Banach spaces of the type C(K)
  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). 
Petr Hajek and Richard Haydon