For every pointed metric space (M,p), the Lipschitz-free space F(M) is the strongly unique isometric predual of operatornameLip_0(M); that is, operatornameLip_0(M)=F(M)^ and no other isometric predual of operatornameLip_0(M) exists in the strong sense of Weaver.

Weaver’s Lipschitz-free-space predual problem

The problem asks whether the natural predual of a Lipschitz space is uniquely determined, in the strong sense proposed by Weaver. The latest preprint claims a complete affirmative theorem, first for length spaces and then for arbitrary metric spaces.

Known results

Weaver, 2018: claimed uniqueness, and in some cases strong uniqueness, for complete convex spaces and finite-diameter spaces.
Weaver’s corrected version withdrew the invalid finite-diameter argument after a flaw in Lemma $3.1$ was identified.
Earlier work established $\operatorname{Lip}_0(X) \cong L^\infty[0,1]$ for separable metric trees, yielding uniqueness in spaces isometrically contained in such trees.
It remains unknown whether a predual can be unique without being strongly unique.

September 2026 claimed solution

Aliaga, Cúth, and Vico claim that Lipschitz-free spaces have strongly unique preduals for arbitrary metric spaces. Their work also gives a counterexample to the earlier assertion that strong uniqueness passes to $1$ -codimensional weak- $\ast$ -closed subspaces. The claim appears in a new unrefereed preprint and has not been independently verified.

Current status (as of September 2026): A complete solution is claimed for arbitrary metric spaces, but the new preprint is unrefereed and the theorem remains unverified; the earlier inheritance lemma is false.

Weaver’s Lipschitz-free-space predual problem

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Additional references

Progress summary

Solutions 0