Smooth and polyhedral approximation in Banach spaces

We show that norms on certain Banach spaces X can be approximated uniformly, and with arbitrary precision, on bounded subsets of X   by C∞C∞ smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on c0(Γ)c0(Γ), where Γ is an arbitrary set. We also give a necessary condition for the existence of a polyhedral norm on a weakly compactly generated Banach space, which extends a well-known result of Fonf.