1 · The essentials
The manuscript separates two ways of approximating a mathematical space by simpler pieces. It constructs a collection of points that can be listed, with a fixed positive minimum distance between distinct points. From these points it builds a space in which vectors can be added and their sizes measured. Within that space, calculations that leave every vector unchanged can be approximated by calculations whose outputs use only finitely many independent directions. These approximations can be made as accurate as desired on each compact set—a set satisfying a precise mathematical condition of limited extent. But there is no single finite ceiling on how much all the required approximations can amplify their inputs. This gives a negative answer to the question attributed to Kalton in the source.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine making simplified copies of a drawing with tools that use only finitely many strokes. You can meet each required accuracy target, but you cannot do so while keeping every tool's sensitivity to its input below one fixed ceiling. The image illustrates the distinction between accuracy and controlled amplification, not the mathematical construction itself.
Why it matters
The example shows that the existence of finite-dimensional approximations does not guarantee uniformly controlled approximations. This distinction persists even when the underlying collection of points is countable and its distinct points are uniformly separated. The manuscript identifies this as a negative answer to Kalton's question.
2 · Going further
The precise result concerns a countable uniformly discrete metric space and its real Lipschitz-free Banach space . Uniform discreteness means that there is a number such that whenever . The construction gives the approximation property while excluding the bounded approximation property.
The approximation property says that, for every compact subset and every , there is a bounded finite-rank linear operator with . The bounded approximation property additionally requires that these choices can all satisfy for one finite constant . Thus the obstruction is not to approximation on compact sets, but to imposing a common bound on the norms of the operators needed for those approximations.
The abstract describes the approach as a construction of a counterexample, but supplies no details of the construction or its argument. Its stated context is Kalton's question, which the example answers negatively. Only the main manuscript is supplied, so no relationship with companion manuscripts can be described. The main result has a Lean formalization; the supplied material does not specify its detailed scope.
Useful notions
- Metric spaces and distances
- Real Banach spaces
- Compact subsets
- Finite-rank linear operators
- Operator norms
What was known before / what is new
Before
The source identifies Kalton's question as the problem addressed by the construction. It provides no account of earlier partial results or of the history of that question.
New
The manuscript constructs a countable uniformly discrete metric space whose real Lipschitz-free space has the approximation property but not the bounded approximation property. Its distinguishing feature is that finite-rank approximation works on every compact set, while no common finite bound on the required operator norms is possible.
Limits and assumptions
The result is an existence statement about a constructed countable uniformly discrete metric space and its real Lipschitz-free Banach space, not a claim about every such space. The main result has a Lean formalization, but the supplied material does not identify the exact scope of that formalization or provide construction details. No companion manuscripts or remaining open questions are specified.
Mini glossary
- Uniformly discrete —
- A metric space is uniformly discrete if one positive number is a lower bound for the distance between any two distinct points.
- Lipschitz-free space —
- The real Banach space associated with the metric space in the manuscript's construction.
- Finite-rank operator —
- A linear operator whose outputs lie in a finite-dimensional subspace.
- Operator norm —
- A measure of the largest factor by which a bounded linear operator can amplify the size of an input.
- Approximation property —
- The property that the identity can be approximated arbitrarily accurately on each compact set by bounded finite-rank linear operators.
- Bounded approximation property —
- The approximation property with the additional requirement that one finite constant bounds the norms of all operators needed for the approximations.
Keywords : Lipschitz-free spaces · Approximation property · Bounded approximation property
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
26/09/2026We 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.