On the unique predual problem for Lipschitz spaces

TL;DR

This paper proves the uniqueness of the predual space for Lipschitz function spaces over various metric spaces, including Banach spaces, highlighting conditions under which this predual is uniquely determined.

Contribution

It establishes the conditions for the uniqueness of the predual of Lip(X) and Lip_0(X), extending known results to broader classes of metric spaces.

Findings

Predual of Lip(X) is unique for any metric space X.

Predual of Lip_0(X) is unique if X has finite diameter or is a Banach space.

Results apply to complete and convex metric spaces, including Banach spaces.

Abstract

For any metric space X, the predual of Lip(X) is unique. If X has finite diameter or is complete and convex --- in particular, if it is a Banach space --- then the predual of Lip_0(X) is unique.