A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

1 paperLean

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.

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▶Introduction: the problem and why it mattersin preparation

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Papers 1

A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

OpenAI · September 26, 2026

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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@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}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

Scope

Kalton's question asks whether uniform discreteness of a metric space forces its Lipschitz-free space to have the bounded approximation property. The formalization constructs a countable metric space with all distinct points at distance at least one whose real Lipschitz-free space has the approximation property but fails every bounded approximation bound. The metric space is unbounded and not proper.

The earlier quantitative renorming result for real \(\ell_1\) is retained: for each integer \(p\ge1\), an equivalent complete norm has the \(2B_p\)-bounded approximation property but fails the \(p\)-bounded approximation property, where \(B_p=300p^2+150p+5\).

Comparator links

Result	Comparator statement
Quantitative renorming of real \(\ell_1\)	RealL1Renorming.lean
Uniformly discrete Lipschitz-free space with AP but no BAP	DiscreteLipschitzFree.lean

Scope note on GitHub →