On the weak ∗ Mazur intersection property and Frechet differentiable norms on dense open sets

Abstract

We prove in this paper the following renorming result: every Banach space with a fundamental biorthogonal system admits an equivalent norm with a dense set of locally uniformly rotund points. Then we obtain new sufficient and necessary conditions for the existence of Fréchet differentiable norms and norms with the Mazur (weak ∗ Mazur) intersection property. These conditions are expressed in terms of norms wich are Fréchet differentiable on dense open sets.