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

TL;DR

This work investigates when the space of Lipschitz functions $\operatorname{Lip}_0(M)$, equipped with its natural $w^*$-topology, is separable for metric spaces $M$ of density at most the continuum. Building on Kalton's framework, it connects $w^*$-separability to embeddings of the Lipschitz-free space $\mathcal F(M)$ into spaces like $c_0(2^\omega)$, yielding several positive cases: Banach spaces with a Markushevich basis, Banach spaces with $w^*$-separable dual unit balls, and complete locally separable metric spaces. The paper also proves that $\operatorname{Lip}_0(M)$ is $w^*$-separable whenever $\operatorname{dens} M \le 2^\omega$ via a constructive, pointwise-separating family of Lipschitz functions, and provides a sufficient condition involving a countable family of 1-Lipschitz functions that guarantees separability in locally separable complete spaces. Collectively, these results advance understanding of the dual structure of Lipschitz-free spaces, offer new reductions for proving $w^*$-separability, and highlight interesting open problems, such as the limits of these reductions and possible universal models under set-theoretic assumptions.

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.

Theorems & Definitions (55)

• Conjecture 1
• Remark 1
• Theorem 2
• Lemma 3
• proof : Proof of Theorem \ref{['thm:kalton11']}
• Corollary 4
• proof
• Corollary 5
• proof
• Theorem 6