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-infinity smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on c(0)(Gamma), where Gamma 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. (C) 2015 Elsevier Inc. All rights reserved.