A pointed metric space is a metric space (M, d) with a chosen basepoint o. For a Lipschitz map f : M -> R, let lip(f) be the least C with |f(x) - f(y)| <= C d(x, y) for all x, y in M. The space Lip_0(M) consists of such real-valued f with f(o) = 0, made a Banach space by the norm lip. The Lipschitz-free space F(M) is the closed linear span of the evaluation functionals delta_x(f) = f(x) inside the dual Lip_0(M)*, so delta_o = 0.
Two properties make F(M) useful. The assignment x -> delta_x is an isometry, and every pointed 1-Lipschitz map u : M -> N induces a contraction F(u) : F(M) -> F(N) that sends delta_x to delta_{u(x)}; metric questions thereby become linear ones. Arens and Eells introduced F(M), and Godefroy and Kalton developed its linear theory.
(Arens and Eells, Pacific J. Math. 6 (1956), 397-403.)
./arens-eells-1956-pjm-embedding-uniform.pdf
(Godefroy and Kalton, Studia Math. 159 (2003), 121-141.)
./godefroy-kalton-2003-studia-lipschitz-free-banach-spaces.pdf
Distance lemma. For all x, y in M, ||delta_x - delta_y||_{F(M)} = d(x, y).
Proof. If lip(f) <= 1 then |f(x) - f(y)| <= d(x, y) for every such f, so ||delta_x - delta_y|| <= d(x, y). For the reverse, g(z) = d(z, y) - d(y, o) has lip(g) <= 1 and g(x) - g(y) = d(x, y). QED
Linearization lemma. A pointed 1-Lipschitz u : (M, d, o) -> (N, rho, o') extends from sum_x c_x delta_x -> sum_x c_x delta_{u(x)} to a unique contraction F(u) : F(M) -> F(N), with F(id) = id and F(v composed with u) = F(v) F(u).
Proof. Take phi in Lip_0(N) with lip(phi) <= 1. The composition phi composed with u lies in Lip_0(M) with lip(phi composed with u) <= 1, so by the duality that defines F(M) and F(N),
< sum_x c_x delta_{u(x)}, phi > = sum_x c_x phi(u(x))
= < sum_x c_x delta_x, phi composed with u >
<= || sum_x c_x delta_x ||.
The supremum over such phi gives || sum_x c_x delta_{u(x)} || <= || sum_x c_x delta_x ||, so the map is well defined on finite evaluation combinations and extends to a contraction. Functoriality holds on evaluations and hence everywhere by continuity. QED
A bounded linear operator is finite-rank when its image is finite-dimensional. A Banach space X has the approximation property (AP) if for every norm-compact C inside X and every eps > 0 there is a finite-rank T : X -> X with sup_{u in C} ||Tu - u|| < eps, the norm of T allowed to depend on C and eps. For lambda >= 1, X has lambda-BAP if such T can always be taken with ||T|| <= lambda. Bounded approximation property (BAP) means lambda-BAP for some finite lambda; metric approximation property (MAP) means 1-BAP. AP says approximation is always possible, while BAP says one bound serves every compact set.
Kalton proved that over every uniformly discrete metric space the free space has AP, and asked whether a uniform bound always exists; the question was restated by Godefroy and Ozawa and in Godefroy's survey. The answer is negative.
(Kalton, Collect. Math. 55 (2004), 171-217.)
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(Godefroy and Ozawa, Proc. Amer. Math. Soc. 142 (2014), 1681-1687.)
./godefroy-ozawa-2014-1201.0847-free-banach-ap.pdf
(Godefroy, Comment. Math. 55 (2015), 89-118, doi:10.14708/cm.v55i2.1104.)
Kalton's own phrasing: a separable Banach space X has the bounded approximation property (BAP) if there is a sequence of finite-rank operators Tn : X -> X such that lim Tn x = x for every x in X; with ||Tn|| <= lambda one says X has lambda-BAP, and the case lambda = 1 is the metric approximation property (MAP).
(Kalton, Collect. Math. 55 (2004), 171-217.)
./kalton-2004-collect-spaces-lipschitz-holder.pdf
Main theorem. There is a countable, unbounded, nonproper pointed metric space (M, d, o) with d(x, y) >= 1 for distinct x, y, such that F(M) has AP but fails lambda-BAP for every finite lambda >= 1.
The space must be nonproper: Dalet proved that over a countable proper metric space F(M) always has MAP. Uniform discreteness cannot be dropped either, since Smith produced countable discrete examples whose free spaces have AP and fail BAP, without a uniform lower bound on the nonzero distances.
(Dalet, Mediterr. J. Math. 12 (2015), 973-986.)
./dalet-2015-1404.3939-free-spaces-proper.pdf
(Smith, Lipschitz-free spaces and Bossard's reduction argument, preprint 2025.)
./smith-2026-2509.00722-lipschitz-free-bossard.pdf
The construction fixes a candidate norm bound p. A bounded space M_p is built with a finite anchor set A_p at which every finite-rank operator of norm at most p moves at least one delta_a, a in A_p, by at least 1/2. This alone settles nothing, since a bounded uniformly discrete space has BAP; the block is a trap at scale p. The blocks are then glued at a common basepoint into a wedge M. Each block is a 1-Lipschitz retract of M, so F(M_p) sits inside F(M) as an isometrically complemented subspace, and a lambda-BAP for F(M) would force lambda-BAP for every F(M_p), contradicting the trap once p >= lambda. Approximation on a compact set never needs more than finitely many blocks, with the BAP of each used separately at a norm that may grow with the set and accuracy; the wedge therefore has AP. The work is in one block.
Block theorem. For each integer p >= 1 there are a bounded countably infinite pointed metric space (M_p, d_p, o) and a finite set A_p inside M_p containing o, with d_p(x, y) >= 1 for distinct x, y and diam(M_p) <= B_p := 300 p^2 + 150 p + 5, such that every bounded finite-rank T_0 : F(M_p) -> F(M_p) with ||T_0|| <= p satisfies max_{a in A_p} ||T_0 delta_a - delta_a|| >= 1/2.
The block has three layers: a sparse Delta-regular colored multigraph whose small neighborhoods mimic a Delta-regular tree; a labeling of each vertex by capped graph distances to the color classes; and a metric that funnels every vertex sharing a label through a single anchor, then joins anchors pairwise. The empirical rooted-neighborhood viewpoint is Benjamini and Schramm's, and the independent-permutation model of random regular graphs is Friedman's.
(Benjamini and Schramm, Electron. J. Probab. 6 (2001), no. 23.)
./benjamini-schramm-2001-distributional-limits.pdf
(Friedman, A proof of Alon's second eigenvalue conjecture and related problems, preprint 2004.)
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For orientation, the empirical rooted-neighborhood notion is due to Benjamini and Schramm: for a sequence of finite connected planar graphs Gj, each with a special vertex called the root chosen uniformly at random among the vertices, the notion of a distributional limit G of such graphs is introduced.
(Benjamini and Schramm, Electron. J. Probab. 6 (2001), no. 23.)
./benjamini-schramm-2001-distributional-limits.pdf
Multiple edges are distinct instances, and each instance has two oriented incidences e and e-bar exchanged by reversal, even for a loop, so a loop contributes two to the degree. Write e : x -> y for initial and terminal vertices, and let D(x) be the set of Delta outgoing incidences at x. A walk is nonbacktracking if it never immediately reverses its preceding incidence; it may revisit vertices and retake loops. A rooted radius-r ball is the induced multigraph on vertices within graph distance r of the root, retaining colors, loops, and multiplicities. Only vertices are points of the metric spaces.
Colored-tree-limit lemma. For every even integer Delta >= 4 and integer k >= 1 there is a deterministic sequence of nonempty finite Delta-regular multigraphs G_m, colored by {1, ..., k}, such that for every fixed integer r >= 0 and every rooted colored radius-r multigraph type tau, the fraction of vertices of G_m whose ball has type tau converges, as m -> infinity, to the probability of tau for the Delta-regular tree with independent uniform vertex colors.
Proof. Fix N and take Delta/2 independent uniform permutations sigma_1, ..., sigma_{Delta/2} of an N-element set. For each a and v create a distinct edge instance joining v to sigma_a(v). Each a gives every vertex one incidence to its image and one to its preimage, so the multigraph is exactly Delta-regular; fixed points give loops and 2-cycles give parallel edges. Color vertices independently and uniformly in {1, ..., k}, independently of the permutations.
Fix r and explore from a root by breadth-first search, revealing each relevant permutation image and preimage at vertices within distance r, and recording without expanding the vertices first met at distance r + 1 so that every edge of the induced radius-r ball is seen. The number of queries and revealed vertices is at most C = C(Delta, r), independent of N. Conditional on the revealed data, each unrevealed part of a permutation is a uniform bijection from unused domain to unused range, so a fresh image or preimage is uniform among at least N - C unused values, at most C of which are already discovered; its chance of hitting a known vertex is at most C / (N - C).
Before any collision, each fresh query adds one vertex and one edge to the exploration tree; reused queries add nothing. A loop or second parallel edge arises exactly when a fresh query hits a known vertex. A union bound over the at most C queries gives collision probability at most C^2 / (N - C) = O_{Delta,r}(N^(-1)). Off this event the radius-r ball is exactly the regular-tree ball. For two distinct roots, run both explorations together with both roots initially discovered; a larger C(Delta, r) bounds the chance of any collision, and then the two balls are disjoint tree balls with independent colors.
Let I_v indicate that the ball at v has type tau, and set X_{tau,N} = (1/N) sum_v I_v and q_tau = P(tree ball has type tau). The one-root exploration gives E X_{tau,N} -> q_tau, and the two-root exploration gives E X_{tau,N}^2 = (1/N) E I_1 + (1 - 1/N) E(I_1 I_2) -> q_tau^2, which also covers non-tree types where q_tau = 0. Hence Var(X_{tau,N}) -> 0, and each frequency converges in probability to q_tau. At radius at most m there are finitely many colored types, so finite union bounds show that for all large N, with positive probability every such frequency is within 1/m of its tree value. Choosing one such realization of size N_m, strictly increasing in m, makes every fixed radius-and-type frequency tend to q_tau. QED
The scalar tests take values in [-L, L], so a Lipschitz inequality holds automatically between opposite values at points separated by at least 2 L. Take the attachment length above 2 L so that no short path crosses a thread. The label slope L / K is made tiny so that comparing across a thread, whose norm error is at most 2 p R + 1, costs less than L. Set
L = 10 p,
R = 3 L = 30 p := R_p,
K = 10 p (R + 1),
D = K + 2 L + 5,
B = 4 R + D,
V = 1 + 4 p B.
Substitution gives B = 300 p^2 + 150 p + 5 = B_p, with R > 2 L and
(2 p R + 1) / K = (60 p^2 + 1) / (300 p^2 + 10 p) <= 1/5,
K + 2 L + 1 = D - 4 < D.
The last inequality leaves room, below the cap D, for a label gap below K, one source-edge step, and one support step of length at most 2 L.
Total variation is bounded by V, and a large degree dilutes the loss from at most one exceptional direction per support point. Choose an even integer Delta >= 4 with V / Delta < 1/20, and set Q = Delta^(2 L + 1) and H = Delta^(D + 1). The geometry gives at most Q nearby successors for a support point and at most H label coordinates below the cap, which bounds the admissible source colors; a walk itself has Delta choices per step. A larger degree helps the averaging but multiplies the walks in the matching count, so after fixing Delta choose an integer k > 2 Delta Q H, making Delta Q H / k < 1/2. Fix a deterministic sequence (G_m)_{m >= 1} from the colored-tree-limit lemma for this Delta and k; all constants and the whole sequence are fixed before any operator appears.
Take disjoint copies of the graphs G_m and call their vertices source vertices. For a source vertex x in G_m define h(x) in {0, ..., D}^k by
h_i(x) = min(D, graph-distance_{G_m}(x, set of color-i vertices)), 1 <= i <= k,
the graph distance being +infinity, hence the coordinate D, if the component of x has no vertex of color i. This uses unit graph distances, before the metric below. For each label v realized by some source vertex, add one new vertex a_v, an anchor, with h(a_v) = v; set A_p = {a_v : v realized} and a(x) = a_{h(x)}. The edges are:
(i) length R from each source vertex x to its anchor a(x),
(ii) length 2 R + ||v - w||_{infinity} between distinct anchors a_v and a_w,
(iii) length 1 on each source edge.
Let M_p be the set of all source vertices and anchors, and let d_p(u, v) be the infimum of lengths of finite paths from u to v, the empty path giving d_p(u, u) = 0. Choose an anchor as basepoint o, which exists because each G_m is nonempty. Each source vertex hangs by a length-R thread from the anchor carrying its label; anchors are joined by edges whose length dominates the sup-distance of the labels. This bounds the space while keeping small balls inside a single source graph.
Geometry lemma. The preceding construction satisfies:
(a) (M_p, d_p, o) is a countably infinite pointed metric space with separation at least one and diameter at most B = B_p. The anchor set A_p is finite and contains o. Each source edge has endpoints at d_p-distance at most one.
(b) The label h is 1-Lipschitz for the sup norm. A source vertex x of color i has h_i(x) = 0, and h(a(x)) = h(x).
(c) For every source vertex x and anchor a, d_p(x, a) >= R and d_p(x, a(x)) = R; in particular d_p(x, o) >= R_p = 30 p.
(d) Every closed ball of radius 2 L has at most Q points, and at most H coordinates of any h(t) are strictly below D.
Proof. (a) There are at most (D + 1)^k labels, so A_p is finite. The disjoint union of the nonempty finite graphs is countable, and adjoining finitely many anchors keeps it countable. The added edges connect all vertices. Every edge length is a positive integer: source and thread lengths are integers, and the sup-distance of integer labels is an integer. For distinct endpoints the lengths of connecting paths form a nonempty set of positive integers, which has a least element, so d_p is attained and is a metric. Distinct points are at distance at least one because every edge has length at least one, and a source edge gives distance at most one between its endpoints. The sup-distance of two labels is at most D, so two source vertices with different anchors join through the two threads and the anchor edge in length at most R + (2 R + D) + R = 4 R + D = B, and with a common anchor in length 2 R; a source vertex and an anchor lie within 3 R + D, two anchors within 2 R + D. The diameter is therefore at most B.
(b) Across a source edge the distances to a fixed color class differ by at most one whenever the color is present in the component, since the edge is appended to a realizing path, and capping at D preserves this; if the color is absent both coordinates equal D. So adjacent source labels differ by at most one in sup norm. Along a thread the labels agree; along an anchor edge the length dominates the sup-distance of the labels. Summing along a shortest path gives ||h(u) - h(v)||_{infinity} <= d_p(u, v). The own-color coordinate is zero by definition, and anchor labels are copied.
(c) A path from a source vertex to an anchor must first enter the anchor set, and the only edges from source vertices to anchors are threads of length R; hence d_p(x, a) >= R. The thread realizes length R, and o is an anchor, so d_p(x, o) >= R.
(d) Every edge other than a source edge has length at least R = 3 L > 2 L, so a shortest path of length at most 2 L uses only source edges. The radius-2 L ball about an anchor is a singleton, and that about a source vertex lies in its graph ball of radius 2 L inside its own G_m. In a Delta-regular multigraph every vertex within graph distance s is reached by a walk of length at most s, and there are at most Delta^t walks of length t from a fixed vertex, so the ball has at most sum_{t=0}^{2L} Delta^t <= Delta^(2 L + 1) = Q points. If h_i(x) < D then x is within graph distance D - 1 of a color-i vertex; witnesses for distinct colors are distinct, so at most sum_{t=0}^{D-1} Delta^t <= Delta^(D + 1) = H coordinates are below D, at source vertices and by copying at anchors. QED
An operator that nearly fixes the anchors yields, at each source vertex, a signed measure of bounded total variation with a large integral against a bump set by the vertex's label. Since finite rank does not mean finite support, a small perturbation first buys a common finite support.
Finite-support lemma. Suppose a bounded finite-rank T_0 : F(M_p) -> F(M_p) satisfies ||T_0|| <= p and ||T_0 delta_a - delta_a|| < 1/2 for every a in A_p. Then there are a finite set S inside M_p containing o and a bounded finite-rank T : F(M_p) -> F(M_p) with T F(M_p) inside span{delta_j : j in S}, ||T|| <= 2 p, and ||T delta_a - delta_a|| < 1 for every a in A_p.
Proof. If T_0 = 0 take T = 0 and S = {o}. Otherwise choose a basis v_1, ..., v_r of the range, with bounded coordinate functionals composed with T_0 giving bounded phi_i and T_0 u = sum_i phi_i(u) v_i. Take 0 < eta < min(p, 1/(2 B)) and finite evaluation combinations w_i with sum_i ||phi_i|| ||w_i - v_i|| < eta, possible by density. Then T u = sum_i phi_i(u) w_i has ||T - T_0|| < eta; let S be {o} together with the supports of the w_i. Then T has the stated support and ||T|| <= p + eta < 2 p. Since ||delta_a|| <= B, ||T delta_a - delta_a|| <= ||T_0 delta_a - delta_a|| + eta B < 1/2 + eta B < 1. QED
For a source vertex x define on all of M_p
b_x(t) = L (1 - ||h(x) - h(t)||_{infinity} / K)_+, s_+ := max(s, 0).
By the geometry lemma the label h is 1-Lipschitz, so b_x is in [0, L], has Lipschitz constant at most L / K <= 1, and equals L at the anchor a(x). For a signed measure nu on a finite set S write |nu|(E) = sum_{j in E} |nu({j})| and ||nu||_{TV} = |nu|(S).
Measures lemma. Let finite S inside M_p contain o, and let bounded T satisfy T F(M_p) inside span{delta_j : j in S}, ||T|| <= 2 p, and ||T delta_a - delta_a|| < 1 for a in A_p. Then there are signed measures (mu_x)_{x in M_p} on S, each of total mass one, with
integral g dmu_x = (T delta_x)(g - g(o)) + g(o)
for every Lipschitz g, with ||mu_x||_{TV} <= V = 1 + 4 p B, and with integral b_x dmu_x >= 3 L / 4 for every source vertex x.
Proof. Write T delta_x = sum_{j in S minus {o}} c_x(j) delta_j; set mu_x({j}) = c_x(j) for j != o and mu_x({o}) = 1 - sum_{j != o} c_x(j). These masses sum to one and give the affine formula.
For the variation, let g equal the sign of mu_x({j}) on S and 0 off S, with sign(0) = 0. Its values lie in [-1, 1], so by the separation assumption lip(g) <= 2, and
||mu_x||_{TV} = integral g dmu_x
= (T delta_x)(g - g(o)) + g(o)
<= 2 ||T delta_x|| + 1
<= 1 + 4 p B,
using ||T|| <= 2 p and ||delta_x|| <= B. For a source vertex x, the geometry lemma gives d_p(x, a(x)) = R, so ||T delta_x - delta_{a(x)}|| <= ||T(delta_x - delta_{a(x)})|| + ||T delta_{a(x)} - delta_{a(x)}|| <= 2 p R + 1. Since b_x(a(x)) = L,
integral b_x dmu_x - L = (T delta_x - delta_{a(x)})(b_x - b_x(o))
>= -(L / K)(2 p R + 1)
>= -L / 5,
because (2 p R + 1) / K <= 1/5. Hence integral b_x dmu_x >= 4 L / 5 >= 3 L / 4. QED
Fix a finite set S inside M_p containing o. Its points are states, and their labels and distances stay fixed for the rest of the argument. The set S is the support produced by the finite-support lemma, and the graph is chosen only after S is fixed. A graph vertex x and a state j in S are compatible when
||h(x) - h(j)||_{infinity} <= K + 2 L + 1 = D - 4.
A nonbacktracking walk x_0, e_1, x_1, ..., e_t, x_t in one graph G_m is matched if there are states j_0, ..., j_t in S such that (x_s, j_s) is compatible for every s and d_p(j_{s-1}, j_s) <= 2 L for 1 <= s <= t. Both the states and the source vertices may repeat; a return along a different parallel edge is allowed, and only the reverse of the preceding incidence is forbidden. The threshold allows for a label gap below K, where a bump is positive, one source-edge step of length at most one, and one state step of length at most 2 L; the inequality K + 2 L + 1 < D also restricts the possible source colors, as used below.
Each state has at most Q possible successors, by the radius-2 L ball bound, while a graph walk meets many possible colors; the ratio between these counts bounds the fraction of roots from which a long matched walk starts.
Rare-walk lemma. For fixed n >= 1, let beta_m be the fraction of vertices of G_m from which a matched walk of length n starts. Then
limsup_{m -> infinity} beta_m <= |S| (H / k) (Delta Q H / k)^n.
Hence one chooses n >= 2 and then m with beta_m < 1 / (20 V).
Proof. For j in S set C_j = {i : h_i(j) < D}, so |C_j| <= H by the geometry lemma. If x has color i and is compatible with j, then h_i(x) = 0 forces h_i(j) <= K + 2 L + 1 = D - 4 < D, so i lies in C_j; this is only a necessary condition, which avoids any independence claim about full labels.
Consider the weaker event that a nonbacktracking length-n walk x_0, ..., x_n from the root has states j_0, ..., j_n with d_p(j_{s-1}, j_s) <= 2 L for 1 <= s <= n and with the color of x_s in C_{j_s} for 0 <= s <= n. This depends only on the induced colored radius-n ball, of which there are finitely many rooted types. With S and n fixed, the sets C_j and the transition relation are fixed, so the event is a fixed finite union of types, and the colored-tree-limit lemma makes its frequency converge to its probability on the independently colored tree. The support depends on the whole graph sequence, but once fixed it only selects types, and no colors are conditioned on it.
Along a uniform nonbacktracking walk with Delta choices per step, there are at most Delta^n paths reaching x_n; a fixed choice of all states gives probability at most (H / k)^(n+1) for the required color memberships, since the vertices of a tree are distinct and overlaps of the C_j do not matter. A union bound gives Delta^n |S| Q^n (H / k)^(n+1) = |S| (H / k) (Delta Q H / k)^n. Every matched walk satisfies the weaker event, proving the display. Finally k > 2 Delta Q H gives Delta Q H / k < 1/2; choose n >= 2 with the right side below 1/(40 V), then m with beta_m < 1/(20 V). QED
Call a vertex of G bad if a matched walk of length n starts there, and good otherwise. At a good vertex the absence of long matched walks picks out at most one direction for each state, which is what makes the construction rigid enough to contradict a bounded operator. For a good vertex x, an incidence e in D(x), and a state j in S, let L_x(e, j) be the set of positive lengths of matched walks starting through e with initial state j, and set r_x(e, j) = max({0} union L_x(e, j)). A matched walk of length at least n has a matched prefix of length n, so L_x(e, j) is inside {1, ..., n - 1} and the maximum is defined; the value 0 says only that no positive-length walk exists.
Call e exceptional at (x, j) if its record is strictly larger than every other outgoing record at (x, j), so there is at most one such incidence; if the largest records are tied, there is none. At a bad vertex, call all outgoing incidences exceptional at every state. For an incidence e : x -> y put E_e = {j in S : e exceptional at (x, j)}.
Exceptional-direction lemma. Let e : x -> y be an incidence and j, j' in S. If (x, j) and (y, j') are compatible and d_p(j, j') <= 2 L, then j lies in E_e or j' lies in E_{e-bar}.
Proof. If either endpoint is bad this is the definition. Suppose both are good and put
A_x = max over g in D(x) minus {e} of r_x(g, j),
A_y = max over f in D(y) minus {e-bar} of r_y(f, j').
The maxima range over nonempty finite sets, since Delta >= 4. If A_y > 0, choose a walk realizing it and prepend e with the state transition j to j'; the old first incidence is not e-bar, so the resulting walk is nonbacktracking and matched. If A_y = 0, the edge alone with states j, j' is a matched walk. The same after reversing e gives
r_x(e, j) >= 1 + A_y, r_y(e-bar, j') >= 1 + A_x.
If neither incidence were exceptional, their records would be at most A_x and A_y, forcing A_x >= 1 + A_y >= 2 + A_x, a contradiction. QED
The proof uses incidence identities and so handles loops and parallel edges. It allows repeated states and vertex revisits, and using the edge alone covers zero records.
For an incidence e : x -> y the bump b_x is prescribed off E_e and minus b_y off E_{e-bar}. Bump bounds control pairs inside one set and far cross pairs at distance at least 2 L; the exceptional-direction lemma controls close cross pairs, at distance below 2 L.
Edge-test lemma. For every incidence e : x -> y of G there is a 1-Lipschitz f_e : M_p -> [-L, L] with f_e(j) = b_x(j) for j in S minus E_e, and f_e(j') = -b_y(j') for j' in S minus E_{e-bar}.
Proof. Within either prescription set the Lipschitz inequality follows from the bump bound. Across the two sets the value difference at j, j' is b_x(j) + b_y(j') <= 2 L, so the inequality holds whenever d_p(j, j') >= 2 L.
Suppose d_p(j, j') < 2 L and b_x(j) > 0. Then ||h(x) - h(j)||_{infinity} < K, and since d_p(x, y) <= 1 and h is 1-Lipschitz,
||h(y) - h(j')||_{infinity}
<= ||h(y) - h(x)||_{infinity} + ||h(x) - h(j)||_{infinity} + ||h(j) - h(j')||_{infinity}
< 1 + K + 2 L.
Thus (x, j) and (y, j') are compatible with state distance at most 2 L, so by the exceptional-direction lemma both prescriptions cannot be present. The same argument applies when b_y(j') > 0. Hence whenever two prescribed states, one from each set, are less than 2 L apart, both prescribed values vanish; taking j = j' makes the data consistent on the overlap of the two prescription sets.
Let P be the union in S and v : P -> [-L, L] the resulting data, which is 1-Lipschitz. If P is empty take f_e = 0. Otherwise the classical scalar Lipschitz extension principle applies, realized for these finite data by an explicit formula followed by clipping:
F(z) = min over j in P of ( v(j) + d_p(z, j) ),
f_e(z) = max(-L, min(L, F(z))).
For z in P the Lipschitz inequalities give F(z) = v(z); the triangle inequality gives F(z) <= F(w) + d_p(z, w) and symmetrically, so F is 1-Lipschitz; and clipping to [-L, L] preserves both the bound and the values. QED
(McShane, Bull. Amer. Math. Soc. 40 (1934), 837-842.)
./mcshane-1934-bams-extension-of-range-of-functions.pdf
Each test compares the signed measures at the two ends of its incidence, losing variation only on the exceptional states. At a good vertex each state is exceptional in at most one of the Delta directions, so averaging the losses against the degree and the bad fraction controls the total.
Proof of the block theorem. For the integer p, the geometry lemma supplies the countably infinite block, the separation and diameter bounds, and the finite anchor set A_p. Suppose a bounded finite-rank T_0 of norm at most p has error less than 1/2 on every anchor. Take T and S from the finite-support lemma and (mu_x) from the measures lemma. With S fixed, choose n >= 2 and then G = G_m by the rare-walk lemma, so that the bad fraction beta is less than 1 / (20 V); then apply the exceptional-direction and edge-test lemmas to these choices.
For an incidence e : x -> y, since f_e - b_x vanishes off E_e and is bounded by 2 L there, and f_e + b_y vanishes off E_{e-bar}, the test prescription gives
integral f_e d(mu_x - mu_y)
>= (3 L / 2) - 2 L (|mu_x|(E_e) + |mu_y|(E_{e-bar})),
regardless of the signs of the coefficients of the measures.
On the other hand, by the affine formula and lip(f_e) <= 1 with values in [-L, L], the two affine pairings combine into a single pairing of delta_x - delta_y:
integral f_e d(mu_x - mu_y) = (T(delta_x - delta_y))(f_e - f_e(o)),
|integral f_e d(mu_x - mu_y)| <= 2 p d_p(x, y) <= 2 p.
The last step uses ||T|| <= 2 p and d_p(x, y) <= 1.
Choose a uniform vertex x of G, then a uniform incidence e in D(x). At a good vertex each state lies in at most one exceptional set, so
(1 / Delta) sum over e in D(x) of |mu_x|(E_e)
<= (1 / Delta) sum over j in S of |mu_x|({j}) <= V / Delta,
while at a bad vertex the average is at most V; hence
E |mu_x|(E_e) <= V / Delta + V beta < 1 / 10.
By Delta-regularity this sampling is uniform over all oriented incidences; reversal preserves it, exchanging the two incidences of a loop as well, so E |mu_y|(E_{e-bar}) < 1 / 10 as well. Taking expectations in the lower bound and using the upper bound forces
2 p > (3 L / 2) - (4 L / 10) = 11 L / 10 = 11 p,
a contradiction since p >= 1. The construction fixed p, all constants, the whole graph sequence, M_p, and A_p before T_0 was considered; only after T_0 produced the finite support S were n and then m chosen, so the same finite anchor set defeats every operator in the class. QED
A bounded uniformly discrete space has a concrete coefficient model and a controlled BAP. The identification with ell_1 is due to Albiac, Ansorena, Cuth, and Doucha, with the specific coefficient bounds stated by Cuth and Kalenda.
(Albiac, Ansorena, Cuth, Doucha, Israel J. Math. 240 (2020), 65-98.)
./albiac-ansorena-cuth-doucha-2020-1811.01265-lipschitz-free-p-spaces.pdf
(Cuth and Kalenda, Lipschitz-free spaces over uniformly discrete metric spaces are 3-Schur, preprint 2026.)
./cuth-kalenda-2026-2604.01875-3-schur.pdf
Coefficient proposition. Let (N, d, o) be a countable pointed metric space with d(x, y) >= 1 for distinct points and diam(N) <= B < infinity. The map J : ell_1(N minus {o}) -> F(N), J c = sum_{x != o} c_x delta_x, is a linear isomorphism onto F(N) with
(1/2) ||c||_1 <= ||J c|| <= B ||c||_1.
If N != {o} then F(N) has 2 B-BAP; if N = {o} both spaces are zero.
Proof. For finitely supported c, ||delta_x|| <= B gives the upper bound. For the lower bound let f_c : N -> R equal sign(c_x)/2 on the support of c off o and 0 elsewhere: it vanishes at o, its values differ by at most one, so by separation it is 1-Lipschitz, and testing against f_c gives ||J c|| >= (J c)(f_c) = (1/2) sum_{x != o} |c_x|.
The upper bound makes J c converge in F(N) for every c in ell_1; both bounds pass to the extension by approximation with finitely supported families. So J is injective with inverse of norm at most two on its range. The range is closed because ||c_m - c_j||_1 <= 2 ||J c_m - J c_j||; it contains every delta_x, whose span is dense, hence it is all of F(N).
For N != {o} (so B >= 1), choose increasing finite E_m exhausting N minus {o} with coordinate projections q_m. Then S_m = J q_m J^{-1} has finite rank and norm at most 2 B, and S_m u -> u for all u because q_m c -> c in ell_1. Convergence is uniform on compact sets: given compact C and eps > 0, take a finite eps/(3 (2 B + 1))-net; for large m the error is below eps/3 on the net, and since ||S_m - id|| <= 2 B + 1 it is below eps on C. If N is finite, eventually q_m = id. The case N = {o} is trivial. QED
Renorming corollary. For each integer p >= 1, real ell_1 carries an equivalent norm || . ||_p with (1/2) ||c||_1 <= ||c||_p <= B_p ||c||_1 such that (ell_1, || . ||_p) has 2 B_p-BAP but fails p-BAP; for p = 1 it fails MAP.
Proof. By the block theorem, M_p minus {o} is countably infinite; enumerate it and let J_p be the coefficient map of the coefficient proposition with B = B_p. Set ||c||_p = ||J_p c||_{F(M_p)}, so J_p is a surjective isometry from (ell_1, || . ||_p) to F(M_p), and the coefficient bounds give the stated norm bounds. Conjugation by J_p preserves norms, finite rank, and approximation errors, so it transfers 2 B_p-BAP to the new norm. If the new norm had p-BAP, conjugation would give p-BAP on F(M_p); applying it with accuracy 1/2 to {delta_a : a in A_p} contradicts the block theorem. For p = 1 this is failure of MAP. QED
The corollary also answers a question of Kalton on equivalent norms on real ell_1: Kalton observed that a positive answer would imply BAP for every uniformly discrete free space, so the case p = 1 already gives a negative answer.
(Kalton, Math. Ann. 354 (2012), 1247-1288.)
./kalton-2012-uniform-structure-of-banach-spaces.pdf
Take disjoint copies of the spaces M_p and identify only their basepoints:
M = {o} union (disjoint union over p >= 1 of M_p minus {o}).
Keep d_p within each block M_p, and for nonbasepoint points in distinct blocks set
d(x, y) = d_p(x, o) + d_q(y, o), x in M_p minus {o}, y in M_q minus {o}, p != q.
Wedge lemma. M is a countable, uniformly discrete pointed metric space with d(x, y) >= 1 for distinct points. It is unbounded and nonproper. For each p there are pointed 1-Lipschitz maps i_p : M_p -> M and r_p : M -> M_p with r_p composed with i_p = identity on M_p.
Proof. Symmetry and positivity are clear. For the triangle inequality, if the three points lie in one block, that block's inequality applies; if the third point z lies in another block, the detour through o has length d_p(x, o) + 2 d_q(z, o) + d_p(y, o) >= d_p(x, y), and a third point equal to o gives equality. So d is a metric. Distinct points are at distance at least one: within a block by the block theorem, and across blocks because each nonbasepoint is at distance at least one from o.
If M had finite diameter C_0, every M_p would have diameter at most C_0, so the coefficient proposition would give 2 C_0-BAP on every F(M_p). For an integer p >= max(1, 2 C_0), applying that property with accuracy 1/2 to {delta_a : a in A_p} would produce a finite-rank operator of norm at most 2 C_0 <= p with error below 1/2 at every anchor, contradicting the block theorem. So M is unbounded.
The closed ball about o of radius B_1 contains the countably infinite 1-separated set M_1; if it were compact, its cover by balls of radius 1/3 would have a finite subcover, each member holding at most one point of M_1, impossible. So M is nonproper.
Let i_p be the inclusion and let r_p : M -> M_p fix M_p and send every other block to o. Both are pointed and 1-Lipschitz: inclusion preserves distances, and for the retraction the only case needing an inequality has x in M_p and y in M_q minus {o} with q != p, where d_p(r_p x, r_p y) = d_p(x, o) <= d(x, y). QED
By the linearization lemma the maps i_p and r_p give contractions I_p = F(i_p) : F(M_p) -> F(M) and P_p = F(r_p) : F(M) -> F(M_p). Since r_p composed with i_p = identity on M_p, one has P_p I_p = identity on F(M_p), and then for every v in F(M_p):
||v|| = ||P_p I_p v|| <= ||I_p v|| <= ||v||,
so I_p is an isometry of F(M_p) onto a 1-complemented subspace of F(M).
Proof of the main theorem. If F(M) had lambda-BAP for a finite lambda >= 1, choose an integer p >= lambda and apply the property with accuracy 1/2 to the finite set {I_p delta_a : a in A_p}, getting U with ||U|| <= lambda. The compression P_p U I_p has finite rank and norm at most lambda <= p, and for every a in A_p:
||P_p U I_p delta_a - delta_a|| <= ||U I_p delta_a - I_p delta_a|| < 1/2,
contradicting the block theorem. All blocks and anchors were fixed before lambda and U were chosen.
For AP, define r_{<= N} : M -> M to keep the first N blocks and send the rest to o; it is pointed and 1-Lipschitz, because distances inside the retained part are unchanged, distances inside the collapsed part become zero, and the cross-block formula bounds the image distance for a retained point and a collapsed point. Its linearization R_N = F(r_{<= N}) = sum_{p <= N} I_p P_p is a contractive projection: on an evaluation delta_x the sum keeps delta_x if x lies in the first N blocks and sends it to 0 otherwise, so R_N fixes every finite evaluation combination once N is large. Such combinations are dense and ||id - R_N|| <= 2, so R_N u -> u for every u; the finite-net argument of the coefficient proposition makes this convergence uniform on compact sets.
Fix compact C inside F(M) and eps > 0. Choose N with sup_{u in C} ||R_N u - u|| < eps/2. For 1 <= p <= N the image P_p(C) is compact, so by the coefficient proposition pick finite-rank T_p : F(M_p) -> F(M_p) with ||T_p|| <= 2 B_p and sup_{u in C} ||T_p P_p u - P_p u|| < eps/(2 N). Then T = sum_{p <= N} I_p T_p P_p has finite rank and bound sum_{p <= N} 2 B_p < infinity, and for u in C:
||T u - u|| <= ||R_N u - u|| + sum_{p <= N} ||T_p P_p u - P_p u|| < eps.
The combined bound may grow with N, which AP permits, so F(M) has AP. With the uniform discreteness, unboundedness, and nonproperness of the wedge lemma, this proves the main theorem. QED
Kalton also asked whether every separable Banach space is approximable: whether there is an equi-uniformly continuous sequence psi_n : X -> X with relatively compact ranges and psi_n(x) -> x pointwise. The point and the net criterion used below are Kalton's, and the space F(M) just constructed is not approximable even though it has AP.
(Kalton, Math. Ann. 354 (2012), 1247-1288.)
./kalton-2012-uniform-structure-of-banach-spaces.pdf
Nonlinear proposition. The space X = F(M) of the main theorem is not approximable.
Proof. The 1-separated set {delta_x : x in M} extends to a maximal 1-separated net N_X inside X, so every point of X lies within distance one of N_X. If X were approximable, one Lipschitz bound Lambda >= 1 would serve for maps g : N_X -> N_X with relatively compact range, and separation makes such a range finite, turning approximation within distance below one into exact fixing.
Fix a finite F inside M containing o and choose g fixing every delta_x, x in F. The restriction x -> g(delta_x) is a pointed Lambda-Lipschitz map M -> X, so by the linearization property of free spaces into a Banach space it extends to a linear T_F : X -> X of norm at most Lambda with T_F delta_x = g(delta_x) for all x. Its range is finite, since the relatively compact range of g lies in the 1-separated set N_X and is therefore finite.
To approximate a compact set, approximate a finite net in it by finite evaluation combinations and put all their supports in F; then T_F fixes those combinations, and ||T_F - id|| <= Lambda + 1 controls the remaining errors. These T_F would be finite-rank operators of uniformly bounded norm approximating on compact sets, that is BAP, contradicting the main theorem. QED
The block rests on classical and modern ingredients. The finite metric-extension lower bounds that separate projections from general operators come from Brudnyi and Brudnyi, and the signed-versus-probability distinction in almost extension is Naor's. The comparison of nonbacktracking continuation lengths that underlies the parent-direction records appears in the work of Ben-David, Childs, Gilyen, Kretschmer, Podder, and Wang, where a consistency lemma excludes an incompatible pair of choices by symmetric length inequalities.
(Brudnyi and Brudnyi, Amer. J. Math. 129 (2007), 217-314.)
./brudnyi-brudnyi-2007-0404304-metric-linear-extensions.pdf
(Naor, Adv. Math. 384 (2021), 107761.)
./naor-2021-2009.11373-almost-extension.pdf
(Ben-David, Childs, Gilyen, Kretschmer, Podder, Wang, SIAM J. Comput. 53 (2024), FOCS20-368--415.)
./ben-david-childs-gilyen-kretschmer-podder-wang-2020-2006.12760-symmetries-graph-properties.pdf