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

What OpenAI claims

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. (Lean)

Paper

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

   26 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's 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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.