Gâteaux differentiability on non-separable Banach spaces

Abstract

This paper deals with the extension of a classical theorem by R. Phelps on the Gâteaux differentiability of Lipschitz functions on separable Banach spaces to the non-separable case. The extension of the theorem is not possible for general non-separable Banach spaces, as shown by Larman. Therefore, we will work on some important non-separable spaces. We consider a norm of the non-separable space $\ell^{\infty}(\mathbb{R})$, showing that it complies with the aforementioned theorem thesis. We also consider other examples as $L^{\infty}(\mathbb{R})$ and $NBV[a,b]$, showing in all cases their sets of $G$-differentiability and some properties of these sets. All of the above, closely following the assumptions in Phelps Theorem, which will extend in the case separable to functions with rank over the Asplund spaces and in the non-separable case for projective limits and locally-Lipschitz cylindrical functions.