On the weak⁎ separability of the space of Lipschitz functions

Abstract

We conjecture that whenever M is a metric space of density at most continuum, then the space of Lipschitz functions is $w^{⁎}$-separable. We prove the conjecture for several classes of metric spaces including all the Banach spaces with a projectional skeleton, Banach spaces with a $w^{⁎}$-separable dual unit ball and locally separable complete metric spaces.