A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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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A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
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…
Full source 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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