Smooth approximations of norms in separable Banach spaces
  Let X be a separable real Banach space having a k-times continuously Fr\'{e}chet differentiable (i.e. C^k-smooth) norm where k=1,...,\infty. We show that any equivalent norm on X can be approximated uniformly on bounded sets by C^k-smooth norms. 
Petr H\'ajek and Jarno Talponen
math.FA math.AG