Lipschitz-free spaces are strongly unique preduals
Abstract
We prove that the Lipschitz-free space ℱ(M) is the strongly unique isometric predual of Lip0(M) for every metric space M, solving a longstanding open problem of Weaver. The result is proved first for length metric spaces by analyzing the behavior of Lipschitz functions on paths in M and the topological properties of subspaces of functions vanishing on certain barrier-sets of M. Then, the result is extended to general metric spaces by embedding them into length spaces in such a way that the restriction operator for Lipschitz functions is weak* continuous with respect to their selected preduals. Finally, we also give a counterexample to the codimension-one inheritance assertion used in Weaver's original proof for bounded and geodesic metric spaces.
Related
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- On the L-embeddability of Lipschitz-free spaces in their biduals
BibTeX
@misc{aliaga2026lipschitzfree,
title = {{Lipschitz-free spaces are strongly unique preduals}},
author = {Ramón J. Aliaga and Marek Cúth and Felipe Vico},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24392},
eprint = {2609.24392},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24392},
}
RIS
TY - UNPB
TI - Lipschitz-free spaces are strongly unique preduals
AU - Ramón J. Aliaga
AU - Marek Cúth
AU - Felipe Vico
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24392
UR - https://arxiv.org/abs/2609.24392
AB - We prove that the Lipschitz-free space ℱ(M) is the strongly unique isometric predual of Lip_0(M) for every metric space M, solving a longstanding open problem of Weaver. The result is proved first for length metric spaces by analyzing the behavior of Lipschitz functions on paths in M and the topological properties of subspaces of functions vanishing on certain barrier-sets of M. Then, the result is extended to general metric spaces by embedding them into length spaces in such a way that the restriction operator for Lipschitz functions is weak^* continuous with respect to their selected preduals. Finally, we also give a counterexample to the codimension-one inheritance assertion used in Weaver's original proof for bounded and geodesic metric spaces.
ER -