On the weak$^*$ separability of the space of Lipschitz functions

TL;DR

This paper investigates the weak$^*$ separability of Lipschitz function spaces, proving the conjecture for specific classes of metric spaces such as Banach spaces with particular properties and locally separable spaces.

Contribution

The paper proves the conjecture that Lipschitz function spaces are $w^*$-separable for several classes of metric spaces, advancing understanding in functional analysis.

Findings

Proved the conjecture for Banach spaces with a projectional skeleton.

Established $w^*$-separability for Banach spaces with a $w^*$-separable dual unit ball.

Confirmed the conjecture for locally separable complete metric spaces.

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.