Embeddability of locally finite metric spaces into Banach spaces is finitely determined

The main purpose of the paper is to prove the following results: Let $A$ be a locally finite metric space whose finite subsets admit uniformly bilipschitz embeddings into a Banach space $X$. Then $A$ admits a bilipschitz embedding into $X$. Let $A$ be a locally finite metric space whose finite subsets admit uniformly coarse embeddings into a Banach space $X$. Then $A$ admits a coarse embedding into $X$. These results generalize previously known results of the same type due to Brown-Guentner (2005), Baudier (2007), Baudier-Lancien (2008), and the author (2006, 2009). One of the main steps in the proof is: each locally finite subset of an ultraproduct $X^\mathcal{U}$ admits a bilipschitz embedding into $X$. We explain how this result can be used to prove analogues of the main results for other classes of embeddings.


Introduction
First we introduce necessary definitions: Definition 1.1. A metric space (A, d A ) is called discrete if there exists a constant δ > 0 such that ∀u, v ∈ A d A (u, v) ≥ δ. A discrete metric space A is called locally finite if for every u ∈ A and every r > 0 the set {a ∈ A : d A (u, a) ≤ r} is finite.
Let C < ∞. A map f : (A, d A ) → (Y, d Y ) between two metric spaces is called .
A map f is called Lipschitz if it is C-Lipschitz for some C < ∞. For a Lipschitz map f we define its Lipschitz constant by .
A map f : A → Y is called a C-bilipschitz embedding if there exists r > 0 such that A bilipschitz embedding is an embedding which is C-bilipschitz for some C < ∞.
The smallest constant C for which there exist r > 0 such that (1) is satisfied is called the distortion of f . A sequence of embeddings is called uniformly bilipschitz if they have uniformly bounded distortions. A map f : (X, d X ) → (Y, d Y ) between two metric spaces is called a coarse embedding if there exist non-decreasing functions ρ 1 , ρ 2 : [0, ∞) → [0, ∞) (observe that this condition implies that ρ 2 has finite values) such that lim t→∞ ρ 1 (t) = ∞ and ∀u, v ∈ X ρ 1 (d X (u, v)) ≤ d Y (f (u), f (v)) ≤ ρ 2 (d X (u, v)). (2) A sequence of embeddings is called uniformly coarse if all of them satisfy (2) with the same ρ 1 and ρ 2 .
The main purpose of this paper is to prove the following two results: Theorem 1.2. Let A be a locally finite metric space whose finite subsets admit uniformly bilipschitz embeddings into a Banach space X. Then A admits a bilipschitz embedding into X.
Theorem 1.3. Let A be a locally finite metric space whose finite subsets admit uniformly coarse embeddings into a Banach space X. Then A admits a coarse embedding into X.
Remark 1.4. It is worth mentioning that our argument implies that a similar result holds for any class E of embeddings provided that (a) There is a notion of being uniformly in E for a collection of maps of finite metric spaces into a Banach space.
(b) The notion in (a) is such that if all finite subspaces of a metric space A admit uniformly-in-E embeddings into a Banach space X, then there is an embedding of the class E of A into an ultraproduct X U where U is a non-trivial ultrafilter (see the construction below).
(c) The image of a locally finite metric space under an embedding of the class E is locally finite.
(d) A composition of an embedding of the class E and a bilipschitz embedding is in E. Our proof uses (a) The way of pasting embeddings of "pieces" suggested in [BL08]. (b) Approaches to selection of basic subsequences developed in [KP65]. (c) Some basic ultraproduct techniques going back to [DK72].

Proof in the bilipschitz case
Proof of Theorem 1.2. We pick a point O in A and let By the assumption there are uniformly bilipschitz maps f i : A i → X. We may and shall assume that f i (O) = 0 and that there is a constant 1 ≤ C < ∞ such that We are going to use some basic facts about ultraproducts of Banach spaces introduced in [DK72]. We refer to [DJT95,Chapter 8] for background on this matter.
Let U be a nontrivial ultrafilter on N. The maps The definition of an ultraproduct immediately implies that f : A → X U is a bilipschitz embedding. Let N = f (A). Since the composition of two bilipschitz embeddings is a bilipschitz embedding, it suffices to find a bilipschitz embedding of N (with the metric induced from X U ) into X.
Note: This passage from A to its image in X U is not essential for the proof of Theorem 1.2, it just simplifies some formulas in our proof. A similar step is more essential for other classes of embeddings. Remark 2.1. We would like to emphasize that the rest of the proof of Theorem 1.2 consists in establishing the fact that a locally finite metric subspace of X U admits a bilipschitz embedding into X.
Observation 2.2. If X is finite-dimensional, then X U is of the same dimension (see [DJT95,Proposition 8.4]), and the proof if completed.
If X = L p (0, 1) for some p ∈ [1, ∞], then each separable subspace of X U is isometric to a subspace of X (see [DJT95,Theorem 8.7] and references in [Ost09,p. 169]), so the proof is completed in this case, too.
In this connection in the rest of the proof we assume that X is infinite-dimensional.
It is clear that N i are finite sets. Using the same argument as in the proof of finite representability of X U in X (see [DJT95,Theorem 8.13]) we get that there exist maps s i : Since the sets N i form an increasing sequence, any subsequence into X and satisfies (3). We are going to construct a bilipschitz embedding of N into X using such subsequences.

Note:
We are going to pass to a subsequence in {s i } ∞ i=1 several times. Each time we keep the notation It is clear that we may assume that X is separable (replacing it by the closure of the linear span of ∞ i=1 s i (N i ), if necessary). For a separable Banach space X there exists a separable 1-norming subspace M ⊂ X * . It can be constructed as follows: is 1-norming. Let M ⊂ X * be a separable 1-norming subspace. Then the natural embedding of X into M * is an isometry. We identify X with its image under this embedding. Since M is separable, there is a subsequence in {s i } such that the sequence {s i (a)} ∞ i=k is convergent in the weak * topology of M * for each a ∈ N k . We denote the weak * limit of this sequence by m(a).
We need to select further subsequences of {s i }. We do this in the following two steps.
Step 1. If for some a, b ∈ N i and some In such a case we assume that for all k > j the condition holds. This goal can be achieved because there are finitely many a, b ∈ N i and because s k (a) converges to m(a) in the weak * topology of M * . Step and select a subsequence satisfying for k ≥ i. This can be achieved because N i is finite and s k (a) converges to m(a) in the weak * topology of M * .
We introduce a map ϕ : A → X by if 2 i−1 ≤ ||a|| ≤ 2 i . One can easily check that the map is well-defined for ||a|| = 2 i .
We start by considering the case where X is isomorphic to its hyperplane, and therefore X is isomorphic to X ⊕ R. In this case we show that the embedding ϕ : A → X ⊕ R given by ϕ(a) = (ϕ(a), ||a||) is a bilipschitz embedding.
The proof in the case when X is not isomorphic to its hyperplane is completed in section 2.4. (We know, by results of [GM93], that spaces which are not isomorphic to their hyperplanes exist.) Now we estimate the Lipschitz constants of ϕ and ( ϕ) −1 . We consider three cases.
2.1 Case 1: In this case we have Using (3) we get and the fact that ϕ is a Lipschitz map is immediate.
The estimate for Lip(( ϕ) −1 ) in the case when ||a|| − ||b|| ≥ 1 1000 ||a − b|| is immediate, we just recall that Hence, to finish the estimate for Lip(( ϕ) −1 ) in Case 1 it remains to consider the case when ||m(a) − m(b)|| < 1 100 ||a − b||. In this case we consider two subcases: We start with subcase (12). Let f i,a,b be the functional found in Step 1. We get Now, as above, we consider the case when ||a|| − ||b|| < 1 1000 ||a − b|| separately, and complete the argument in the same way as above.
We turn to the subcase (13). Recall (see (3) Now, as it was done already twice, we consider the case when ||a|| − ||b|| < 1 100 ||a − b|| separately, and complete the argument in the same way as above. This completes the argument in Case 1.

Case 2:
We have Estimate from above: The first and the last terms have norms ≤ 4(||a||−||b||). The norm of the two remaining terms can be estimated as follows: (16) Now we turn to estimates from below. Rewriting and estimating some of the terms as in (16) we get where in the last line we used (3). We complete the proof in this case as three times before. If ||a|| − ||b|| < 1 20 ||a − b||, we get an estimate from (17). Otherwise we use (11).

Case 3:
In this case we have

Completion of the proof for spaces non-isomorphic to their hyperplanes
We have proved Theorem 1.2 in the case when X is isomorphic to its hyperplane. To prove Theorem 1.2 in the general case we find a Lipschitz map τ : R + → X such that the map ϕ : N → X given by ϕ(a) = τ (||a||) + ϕ(a) works just in the same way as ϕ. It is easy to see that for this to be true we need the inequality to hold for some α > 0. We rewrite this inequality as It is clear that all these sets are finite. We construct inductively a sequence {F i } ∞ i=1 of finite-dimensional subspaces of X and a sequence {p i } ∞ i=1 of vectors. We let F 1 = lin(T 1 ). Since X is infinitedimensional (see the assumption made after Observation 2.2), there is p 1 ∈ S X such that dist(p 1 , F 1 ) = 1. Let F 2 = lin(T 2 ∪ {p 1 }) and p 2 ∈ S X be such that dist(p 2 , F 2 ) = 1. Let F 3 = lin(T 3 ∪ {p i } 2 i=1 ), we continue in an obvious way. We introduce the map τ : R + → X in the following way: Since ||p i || = 1, the map τ is 1-Lipschitz. It remains to show that the inequality (18) holds. We consider three cases: 1. 3 i ≤ ||b|| ≤ ||a|| ≤ 3 i+1 . The argument used in this case can be used also in the case 0 ≤ ||b|| ≤ ||a|| ≤ 3 1 . Minor adjustments of the other cases are needed if 0 ≤ ||b|| ≤ 3 1 ≤ ||a||.
The last inequality follows from dist(p i+1 , F i+1 ) = ||p i+1 || and ϕ(a), In the second case we consider two subcases: In subcase (19) we get In subcase (20) we have In this chain of inequalities we use the fact that ϕ(a), ϕ(b) ∈ T i , dist(p i , F i ) = ||p i ||, in the last line we use the inequality ||b|| ≤ 3 i ≤ ||a|| and (20). Now we consider the third case. In this case we again consider subcases (19) and (20). In the first subcase the argument is exactly as above. So we focus on the second subcase. In this subcase we have see the defining inequality for the third case.
In this subcase we have where r is a vector contained in F i . Thus where we use the fact that r and ϕ(a) − ϕ(b) are in F i and dist(p i , F i ) = ||p i ||. This completes the proof of (18) and thus Theorem 1.2.

Proof in the coarse case
Proof of Theorem 1.2 contains almost everything we need for the proof of Theorem 1.3, we need just to modify the beginning of the proof.
The definition of an ultraproduct immediately implies that f : A → X U is a coarse embedding. Let N = f (A), it is easy to check that N with the metric induced from X U is a locally finite metric space. The argument of the proof of Theorem 1.2 shows that there is a bilipschitz embedding of N into X (see Remark 2.1). Since the composition of a coarse and a bilipschitz embeddings is a coarse embedding, the proof is completed.

Relations with previous results
The main result of the paper [BL08] is Theorem 4.1. If X is a Banach space without cotype, then every locally finite metric space admits a bilipschitz embedding into X.
The fact that Theorem 4.1 implies the following result of [BG05] was observed already in [BL08].
Theorem 4.2 ( [BG05]). Let A be a metric space with bounded geometry. There exists a sequence of positive real numbers {p n } and a coarse embedding of A into the ℓ 2 -direct sum of {ℓ pn } ∞ n=1 . In [Ost09] the following result was proved Theorem 4.3. Let A be a locally finite metric space which admits a bilipschitz embedding into a Hilbert space, and let X be an infinite-dimensional Banach space. Then there exists a bilipschitz embedding f : A → X.
In [Ost06b] a coarse version of this result was proved. These results follow immediately from Theorems 1.2 and 1.3 and the Dvoretzky theorem [Dvo61].
Theorem 1.2 can also be used to derive both of the main results of [Bau07] from the mentioned in [Bau07] finite versions of the results.