Discrete distances do not guarantee bounded approximation

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Family 330 · Functional analysis
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

A discrete metric space may seem easy to approximate: distinct points stay apart, so there are no arbitrarily small geometric features to resolve. But linearizing its geometry can produce a Banach space with much subtler approximation behavior.

A Lipschitz-free space is a canonical linear space built from a metric space. It preserves the original distances between designated point vectors and turns Lipschitz maps into linear operators. Kalton asked whether uniform separation of the original points guarantees that this free space has the bounded approximation property.

Family 330 claims a counterexample. It constructs a countable metric space in which every two distinct points are at least one unit apart, but the associated free space cannot approximate its identity using finite-rank operators with one common norm bound.

Ordinary approximation still works. Every compact set can be approximated as accurately as desired by some bounded finite-rank operator. The obstruction is the need to let the operator norm grow as the approximation tasks become more demanding.

2. Background and essential definitions

A pointed metric space $(M,d,o)$ is a metric space with a selected basepoint $o$. Let $\operatorname{Lip}_0(M)$ be the real-valued Lipschitz functions vanishing at $o$, with norm

$$\|f\|_{\mathrm{Lip}} =\sup_{x\ne y}\frac{|f(x)-f(y)|}{d(x,y)}.$$

Each point $x$ defines an evaluation functional $\delta_x(f)=f(x)$. The real Lipschitz-free space $\mathcal F(M)$ is the closed linear span of these evaluations inside the dual of $\operatorname{Lip}_0(M)$. Its norm satisfies

$$\|\delta_x-\delta_y\|=d(x,y), \qquad \delta_o=0.$$

Thus the original metric embeds isometrically in a linear space. A basepoint-preserving Lipschitz map into a Banach space has a corresponding bounded linear map on the free space, with the same norm as its Lipschitz constant. This is the construction’s linearization principle.

A Banach space has the approximation property, AP, if its identity can be approximated uniformly on every norm-compact set by bounded finite-rank linear operators. It has $\lambda$-BAP if these operators can always be chosen with norm at most $\lambda$. BAP means $\lambda$-BAP for some finite $\lambda$, and the metric approximation property, MAP, means 1-BAP.

Uniform discreteness means that all nonzero distances have one common positive lower bound. This is stronger than merely requiring every point to be isolated. A metric space is proper when every closed bounded ball is compact. A bounded infinite uniformly discrete set is not compact, so uniform discreteness alone does not imply properness.

3. Development history and previous partial results

The free-space construction goes back to the molecular spaces of Arens and Eells. Godefroy and Kalton developed its role in relating Lipschitz geometry to linear Banach-space structure. It provides a setting where a question about distances can become a question about operators.

Kalton established AP for free spaces over uniformly discrete metric spaces and asked whether BAP always follows. Godefroy and Ozawa restated the question in their work on free spaces and approximation properties; it also appears in Godefroy’s survey.

Several positive and negative results approached the boundary. Dalet proved MAP for free spaces over countable proper metric spaces. For bounded uniformly discrete spaces, the free space is isomorphic to $\ell_1$, the space of absolutely summable sequences, and therefore has BAP. These cases leave room for an unbounded, nonproper example assembled from increasingly difficult bounded pieces.

Smith constructed countable complete discrete spaces whose free spaces have AP but fail BAP. M1 emphasizes that these examples do not assert one uniform positive lower bound on all distances. The proposed new example supplies exactly that stronger condition.

Cúth and Kalenda’s April 2026 work proves a quantitative Schur property for uniformly discrete free spaces: weakly convergent sequences must converge in norm, with a quantitative strengthening. This is substantial sequential regularity, yet their introduction still records the BAP question. M1 claims that this regularity and ordinary approximation can coexist with failure of every bounded approximation constant.

4. What the new manuscript claims

M1 constructs a countable, unbounded, nonproper pointed metric space $M$ satisfying

$$d(x,y)\geq1\qquad(x\ne y)$$

such that $\mathcal F(M)$ has AP but fails $\lambda$-BAP for every finite $\lambda\geq1$.

The main local ingredient is more quantitative. For each integer $p\geq1$, M1 constructs a bounded countably infinite metric block $M_p$ and a finite set of anchors $A_p$, including the basepoint. Distinct points remain at least one apart, and

$$\operatorname{diam}(M_p)\leq B_p, \qquad B_p=300p^2+150p+5.$$

Every finite-rank operator $T$ on $\mathcal F(M_p)$ with norm at most $p$ must make a visible error at an anchor:

$$\max_{a\in A_p}\|T\delta_a-\delta_a\|\geq\frac12.$$

Each block nevertheless has $2B_p$-BAP. Thus a block defeats a prescribed small bound without losing all bounded approximation. Joining all the blocks defeats every finite bound simultaneously.

The block also yields an equivalent norm on real $\ell_1$ with $2B_p$-BAP but without $p$-BAP. In the manuscript’s coefficient coordinates,

$$\frac12\|c\|_1\leq\|c\|_p\leq B_p\|c\|_1.$$

For $p=1$, this gives a renorming of $\ell_1$ without MAP.

Finally, M1 derives a nonlinear consequence through an existing theorem of Kalton: the separable space $\mathcal F(M)$ is not approximable in Kalton’s uniform sense. There is no equi-uniformly continuous sequence of maps with relatively compact ranges converging pointwise to its identity. Equi-uniform continuity requires one continuity bound to work for every map in the sequence.

5. Why the advance matters

Importance: 5/5. The claimed construction answers Kalton’s uniformly discrete free-space question negatively and separates several approximation principles that might otherwise appear closely tied.

It shows why ordinary AP is weaker than BAP in a concrete metric setting. At every finite stage, approximation is possible. What fails is uniform control across all stages. The metric space has no arbitrarily small distances, so this failure must arise from how large-scale pieces interact with linear approximation.

The result also warns against confusing an obstruction to a special projection with an obstruction to all finite-rank approximations. Earlier finite metric extension problems can force large projection norms onto a prescribed range. But in a finite-dimensional free space, the identity itself is already a norm-one finite-rank operator. M1’s infinite blocks aim at the stronger obstruction needed here.

The nonlinear consequence adds another distinction: linear AP does not by itself imply uniformly controlled nonlinear approximation by compact-range maps. These implications concern different quantifiers and different kinds of control, even though all describe ways to approximate the identity.

6. The main idea

The global assembly is straightforward once the blocks exist. Identify their basepoints and define the distance between points in different blocks as the sum of their distances to the common basepoint. This produces a metric wedge.

Each block has a 1-Lipschitz retraction from the wedge: keep its points and send the other blocks to the basepoint. Linearization gives contractive inclusion and retraction maps between the corresponding free spaces. Any BAP bound for the whole space would therefore work for every block, contradicting the obstruction for sufficiently large $p$.

AP survives for a different reason. On a fixed compact set, first retain only finitely many blocks. Then approximate within each retained block using its own finite BAP constant. Only finitely many constants occur in this particular task, so their maximum is finite. AP permits that maximum to grow for later tasks.

The difficult work lies inside a block. M1 uses a sequence of finite colored graphs and attaches their vertices to finitely many anchors. Labels record capped distances to color classes. Long attachment edges keep the whole block bounded while preserving useful local graph geometry.

Suppose a small-norm finite-rank operator approximates all anchor evaluations. A small perturbation lets its range be supported on one finite subset of the metric block. Its values can then be represented by signed measures on that support, after adding a basepoint coefficient to normalize total mass. The measures need not be positive.

Lipschitz bump functions near the anchor labels force these signed measures to give large pairings with certain tests. On the other hand, a long nonbacktracking walk in a source graph is unlikely to be tracked by nearby support points with matching labels. Nonbacktracking means that a step never immediately reverses the preceding edge.

At most one direction can be exceptional for each support point in the relevant local comparison. This permits opposing scalar tests at the two ends of an edge. A scalar Lipschitz extension principle makes the tests globally defined, and averaging over many graph directions keeps the exceptional contribution small.

The resulting lower bound on an averaged pairing conflicts with the operator-norm upper bound on adjacent vertices. Importantly, the entire graph sequence and anchor set are fixed first. The operator and its support come later, after which the argument chooses a sufficiently large graph. This order is necessary to obstruct every candidate finite-rank operator, rather than only a preselected collection.

7. Scope, limitations, and open questions

The example is uniformly discrete and countable, but unbounded and nonproper. It does not contradict the bounded uniformly discrete case or Dalet’s theorem for countable proper spaces. The local blocks retain BAP; failure arises from their unbounded sequence of required constants.

The scalar field in the manuscript is real. The principal conclusion is failure of every uniform finite bound, not failure of AP. The additional nonlinear statement uses Kalton’s specific definition involving equi-uniform continuity and relatively compact ranges.

The construction does not classify all uniformly discrete spaces whose free spaces have BAP or supply optimal constants for the individual blocks. Finding simpler examples, sharper growth of the block constants, and geometric conditions guaranteeing BAP remain separate questions. The results and the construction mechanism are reported as stated in M1.

8. Sources and further reading

• Godefroy and Ozawa, Free Banach spaces and the approximation properties.
• Dalet, Free spaces over some proper metric spaces.
• Smith, Lipschitz-free spaces and Bossard’s reduction argument.
• Cúth and Kalenda, Lipschitz-free spaces over uniformly discrete metric spaces are 3-Schur.

Source manuscripts

• M1. OpenAI, A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces (September 26, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.