Gâteaux or Fréchet differentiable norms on duals of interpolation spaces

Abstract

AbstractLet be an interpolation couple of Banach spaces, let , and let be the corresponding real interpolation space. We consider a norm on the space , equivalent to the usual interpolation norm, and we show that if the norms of the dual spaces  and  are Gâteaux (resp., Fréchet) differentiable, then the dual norm on has the same property. If we only assume that one of the two duals is Gâteaux (or Fréchet) differentiable, we are able to show that there is an equivalent Gâteaux (or Fréchet) differentiable dual norm on the space .