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A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
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Abstract
We construct a countable uniformly discrete metric space whose real Lipschitz-free space has the approximation property but fails the bounded approximation property, answering Kalton's question negatively. The identity can be approximated on every compact set by finite-rank operators, but no uniform bound on their norms is possible.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for result 330 (lean/docs/330.md, October 2026). The release's own Lean scope document lists two comparator statements:
DiscreteLipschitzFree.leanfor Theorem 1.1 (a uniformly discrete space whose free space has AP but fails BAP) andRealL1Renorming.leanfor the quantitative renorming of real $\ell_1$ (Corollary 1.3). [family-level]
Possibly relevant literature (found by the agent, not checked)
- B. M. Braga, G. Lancien, C. Petitjean, A. Procházka, On Kalton's interlaced graphs and nonlinear embeddings into dual Banach spaces (arXiv:1909.12132, 2019; v2 2021). Remark 6.4 records Kalton's open problem of whether $\mathcal{F}(M)$ has MAP for every bounded uniformly discrete $M$, which bears on the paper's case $p=1$ of Theorem 1.2 and Corollary 1.3.
- G. Godefroy, Extensions of Lipschitz functions and Grothendieck's bounded approximation property (North-Western European Journal of Mathematics 1 (2015), 29–36). For compact $M$, it characterizes BAP of $\mathcal{F}(M)$ by uniform (near-)extension of Lipschitz functions from finite nets, relevant to the paper's comparison between extension-operator obstructions and its finite-anchor obstruction.
- M. Doucha, P. L. Kaufmann, Approximation properties in Lipschitz-free spaces over groups (arXiv:2005.09785, 2020; v2 2021). It gives positive results (MAP or Schauder bases) for free spaces over certain compact and uniformly discrete group metrics, a contrasting class to the paper's counterexample in Theorem 1.1.
Digests and expositions
- Latent Seminar, Uniformly discrete free spaces with AP but no BAP (Latent Seminar, 2026-10-06). Explains the finite-anchor obstruction, signed-measure tests, wedge construction and nonlinear consequence, stating that it does not independently verify the complete proofs. [family-level]
- Claude Opus 5.5 / Open Mathematics Courses, Lipschitz-free spaces and bounded approximation (Open Mathematics Courses, undated). Five lessons with exercises on the paper's bounded blocks, colored graphs, signed measures, matched walks and scalar tests; the source describes itself as model-written with independent review incomplete. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 330
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026,
author = {{OpenAI}},
title = {{A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026/main.pdf}{OAI:Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/330.md. The Lean statements have not been checked by people against the paper's claims.
Family 330: A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively. Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound.
This is the only paper in this family.