# Some remarks on the structure of Lipschitz-free spaces.

1 Introduction

Let (M, d) be a metric space and 0 [member of] M be a distinguished point. The triple (M, d, 0) is called pointed metric space. By [Lip.sub.0](M) we denote the Banach space of all Lipschitz real valued functions f : M [right arrow] R, such that f (0) = 0. The norm of f [member of] [Lip.sub.0](M) is defined as the smallest Lipschitz constant L = Lip(f) of f, i.e.

[mathematical expression not reproducible]

The Dirac map [delta] : M [right arrow] [Lip.sub.0](M)* defined by <f, [delta](p)> = f(p) for f [member of] [Lip.sub.0] (M) and p [member of] M is an isometric embedding from M into [Lip.sub.0] (M)*. Note that [delta](0) = 0. The closed linear span of {[delta](p), p [member of] M} is denoted F(M) and called the Lipschitz-free space over M (or just free space, for short). Clearly,

[||m||.sub.F(M)] = sup {<m, f> : f [member of] [Lip.sub.0](M), ||f|| [less than or equal to] 1}

It follows from the compactness of the unit ball of [Lip.sub.0](M) with respect to the topology of pointwise convergence, that F(M) can be seen as the canonical predual of [Lip.sub.0](M), i.e. F(M)* = [Lip.sub.0](M) holds isometrically ( Chapter 2 for details).

Lipschitz free spaces have gained importance in the non-linear structural theory of Banach spaces after the appearance of the seminal paper  of Godefroy and Kalton, and the subsequent work of these and many other authors e.g. , , , , , , , , , , , , , ,  , , . Free spaces can be used efficiently for constructions of various examples of Lipschitz-isomorphic Banach spaces X, Y which are not linearly isomorphic. To this end, structural properties of their free spaces F(X), as well as free spaces of their subsets, enter the game. For example, in the separable setting, F(X) contains a complemented copy of X , and it is isomorphic to its [l.sub.1]-sum. On the other hand, if N is a net in X then F(N) is a Schur space  and it has the approximation property.

A comprehensive background on free spaces of metric spaces can be found in the book of Weaver . There are several surveys exposing the applications of this notion to the nonlinear structural theory of Banach spaces, in particular , .

Our first observation in this note is that F(M) contains a complemented copy of [l.sub.1] ([GAMMA]), where [GAMMA] is the density character of an arbitrary infinite metric space M. Our proof could be adjusted also to the case [GAMMA] = [[omega].sub.0], which is one of the main results in .

The main purpose of this note is to prove several structural results, focusing mainly on the case when M is a uniformly discrete metric space, in particular a net N in a Banach space X. Our results run parallel (as we have realized during the preparation of this note) to those of Kaufmann , resp. Dutrieux and Ferenczi  which are concerned with the bigger (in a sense) space F(X). However, the space F(N) is only the linear quotient of F(X), so the results are certainly not formally transferable. In particular, the discrete setting prohibits the use of the "scaling towards zero" arguments (used e.g. in ), which leads to complications in proving that our free spaces are linearly isomorphic to their squares, or even [l.sub.1]-sums. We are able to show these facts at least for nets in finite dimensional Banach spaces and all classical Banach spaces. Surprisingly, the proofs for the finite dimensional case and the infinite dimensional case are rather different.

Our main technical result is that F(N) has a Schauder basis for all nets in C(K) spaces, K metrizable compact. The constructive proof is obtained in [c.sub.0], and the result is then transferred into the C(K) situation by using the abstract theory developed in the first part of our note.

Let us start with some definitions and preliminary results. Let N [subset] M be metric spaces, and assume that the distinguished point 0 [member of] M serves as a distinguished point in N as well. Then the identity mapping leads to the canonical isometric embedding F(N) [??] F(M) ( p.42). In order to study the complementability properties of this subspace, one can rely on the theory of quotients of metric space, as outlined in  p.11 or . For our purposes we will outline an alternative (but equivalent) description of the situation.

Definition 1. Let N [subset] M be metric spaces, 0 [member of] N. We denote by

[Lip.sub.N](M) = {f [member of] [Lip.sub.0] (M) : f [|.sub.N] = 0}.

It is clear that [Lip.sub.N] (M) is a closed linear subspace of [Lip.sub.0] (M), which is moreover w (*)-closed. Indeed, by the general perpendicularity principles ( p.56) we obtain

[Lip.sub.N](M) = F[(N).sup.[perpendicular to]], F(N) = [Lip.sub.N][(M).sub.[perpendicular to]]

Hence there is a canonical isometric isomorphism

[Lip.sub.N] (M) [congruent to] (F(M)/F(N))*

Since the space of all finite linear combinations of Dirac functionals is linearly dense in F(M), resp. also in F(N), it is clear that the image of finite linear combinations of Dirac functionals supported outside the set N, under the quotient mapping F(M) [right arrow] F(M)/F(N) is linearly dense. Moreover, it is nonzero for nontrivial combinations.

Definition 2. If [mathematical expression not reproducible] then we let

[mathematical expression not reproducible]

i.e. we complete the space of finite sums of Dirac functionals with respect to the duality

<[F.sub.N](M), [Lip.sub.N](M)>.

Clearly, our definition gives an isometric isomorphism

[F.sub.N](M) [congruent to] F(M)/F(N)

Proposition 1. Let N [subset] M be metric spaces. If there exists a Lipschitz retraction r : M [right arrow] N then

F(M) [congruent to] F(N) [direct sum] [F.sub.N](M).

This follows readily from the alternative description using metric quotients (e.g. in  Lemma 2.2) using the fact that [F.sub.N](M) [congruent to] F(M/N).

We say that the metric space (M, d) is [delta]-uniformly discrete if there exists [delta] > 0 such that d(x, y) [greater than or equal to] [delta], x, y [member of] M. The metric space is uniformly discrete if it is [delta]-uniformly discrete for some [delta] > 0.

If [alpha], [beta] > 0 we say that a subset N [subset] M is a ([alpha], [beta])-net in M provided it is [alpha]-uniformly discrete and d(x, N) < [beta], x [member of] M.

It is easy to see that every maximal [delta]-separated subset N [subset] M, which exists due to the Zorn maximal principle, is automatically a ([delta], [delta] + [epsilon])-net, for any [epsilon] > 0.

Proposition 2. Let (M, d, 0) be a pointed metric space, K > 0, [{[M.sub.[alpha]]}.sub.[alpha][member of][GAMMA]] be a system of pairwise disjoint subsets of M, and 0 [member of] N [subset] M [[union].sub.[alpha][member of][GAMMA]][M.sub.[alpha]]. Suppose that for all [beta] [member of] [GAMMA] and all x [member of] [M.sub.[beta]] holds

d(x, [U.sub.[alpha][member of][GAMMA],[alpha]/[beta]][M.sub.[alpha]]) [greater than or equal to] Kd(x, N).

Then

[mathematical expression not reproducible]

In particular, if N = {0} then

[mathematical expression not reproducible]

Proof. The result is immediate as any collection of 1-Lipschitz functions [f.sub.[alpha]] [member of] [Lip.sub.N](N [union] [M.sub.[alpha]]) is the restriction of a [1/K]-Lipschitz function [Florin] [member of] [Lip.sub.N](N [union] [[union].sub.[alpha][member of][GAMMA]][M.sub.[alpha]])

Recall that the density character dens(M), or just density, of a metric space M is the smallest cardinal [GAMMA] such that there exists dense subset of M of cardinality [GAMMA].

Let [GAMMA] be a cardinal (which is identified with the smallest ordinal of the same cardinality). By the cofinality cof ([GAMMA]) we denote the smallest ordinal [alpha] (in fact a cardinal) such that [GAMMA] = [lim.sub.[beta]<[alpha]] [[GAMMA].sub.[beta]], where [[GAMMA].sub.[beta]] is an increasing ordinal sequence ( p.26).

2 Structural properties

Proposition 3. Let M be a metric space of density dens(M) = [GAMMA]. Then F(M) contains a complemented copy of [l.sub.1] ([GAMMA]).

Proof. For convenience we may assume that [GAMMA] > [[omega].sub.0], because this case has been already proved in  (Our proof can be adjusted to this case as well). By ( Corollary 1.2) if [c.sub.0]([GAMMA]) [??] X* then [l.sub.1] ([GAMMA]) is complemented in X. So it suffices to prove that [Lip.sub.0](M) contains a copy of [c.sub.0]([GAMMA]). For every n [member of] N let [M.sub.n] be some maximal [1/[2.sup.n]]-separated set in M. Denote [[GAMMA].sub.n] = |[M.sub.n]|. It is clear that dens(M) = [lim.sub.n[right arrow][infinity]] [[GAMMA].sub.n], in the cardinal sense. In case when the cofinality cof([GAMMA]) > [[omega].sub.0], it is clear that [[GAMMA].sub.n] = [GAMMA], for some n [member of] N. In this case, let {[f.sub.[alpha]] : [alpha] [member of] [[GAMMA].sub.n]} be a transfinite sequence of 1-Lipschitz functions such that [[Florin].sub.[alpha]]([x.sub.[alpha]]) = [1/[2.sup.n+3]] and supp ([[Florin].sub.[alpha]]) [subset] B([x.sub.[alpha]], [1/[2.sup.n+3]]). Since the supports of [[Florin].sub.[alpha]] are pairwise disjoint it is clear that [mathematical expression not reproducible] is equivalent to the unit basis of [c.sub.0] ([GAMMA]) and the result follows. In the remaining case, we may assume that [mathematical expression not reproducible] is a strictly increasing sequence of cardinals. Denote [mathematical expression not reproducible]. Let [L.sub.1] = [M.sub.1]. By induction we will construct sets [L.sub.n] [subset] [M.sub.n] as follows. Inductive step towards n + 1. Consider the sets

[mathematical expression not reproducible]

If there is some j, [alpha] so that [mathematical expression not reproducible] then we let [L.sub.n+1] = [A.sub.j,[alpha]]. Otherwise we let

[mathematical expression not reproducible]

In either case we have [mathematical expression not reproducible]. By discarding suitable countable subsets of these sets [L.sub.n] we can assume that

[mathematical expression not reproducible]

To finish, let {[f.sup.n.sub.[alpha]] : [x.sup.n.sub.[alpha]] [member of] [L.sub.n], n [member of] N} be a transfinite sequence of 1-Lipschitz disjointly supported functions such that [mathematical expression not reproducible] and supp [mathematical expression not reproducible]. This sequence is equivalent to the basis of [c.sub.0]([GAMMA]), which finishes the proof.

Theorem 4. Let N, M be uniformly discrete infinite sets of the same cardinality such that N [subset] M is a net. Then F(N) [congruent to] F(M).

Proof. Let K > 0 be such that [max.sub.m[member of]M] dist(m, N) [less than or equal to] K Let r : M [right arrow] N be a retraction such that d(x, r(x)) [less than or equal to] K. As M is uniformly discrete, r is Lipschitz. By Proposition 1

F(M) [congruent to] F(N) [direct sum] [F.sub.N] (M).

It is clear that [F.sub.N](M) [congruent to] [l.sub.1] (M \ N). By Proposition 3

F(N) [congruent to] F(N) [direct sum] [l.sub.1] (M) [congruent to] F(M).

Recall that all nets in a given infinite dimensional Banach space are Lipschitz equivalent (, or  p.239), hence their free spaces are linearly isomorphic. On the other hand, there are examples of non-equivalent nets in [R.sup.2] (,  or  p.242), hence the next result is not immediately obvious.

Proposition 5. Let N, M be nets of the same cardinality dens(M) in a metric space (M,d). Then F(N) [congruent to] F(M).

Proof. Suppose N is a (a, b)-net and M is a (c, d)-net in M, a [less than or equal to] c. Let K = M [union] N, and let K [subset] K be maximal subset such that from each pair of points x [member of] M,y [member of] N for which d(x,y) < [a/4] we choose only one x [member of] K. It is now clear that both N and M are bi-Lipschitz equivalent to a respective subset of K. By Theorem 4, F{K) [congruent to] F(M) [congruent to] F(N).

Of course, the above proposition applies to any pair of nets in a given Banach space X, or its subset S [subset] X which contains arbitrarily large balls.

Lemma 6. Let Y = X [direct sum] R be Banach spaces, N be a net in X and M be the extension of N into the natural net in Y. Denote [M.sup.+] = M [intersection] X [direct sum] [R.sup.+], [M.sup.-] = M [intersection] X [direct sum] [R.sup.-].

If F(N) = F(N) [direct sum] F(N) and F([M.sup.+]) = F([M.sup.+] ) [direct sum] F([M.sup.+]) then F(M) = F([M.sup.+]) = F(M) [direct sum] F(M).

Proof. Thanks to Proposition 5 we are allowed to make additional assumptions on the form of the nets. Let us assume that M = N x Z, which immediately implies that N [union] [M.sup.+] is bi-Lipschitz equivalent with [M.sup.+] (and [M.sup.-]) by translation. Denoting P : Y [right arrow] X the canonical projection P(x, t) = x, we see that P : M [right arrow] N is a Lipschitz retraction, so

F([M.sup.+]) [congruent to] F(N [union] [M.sup.+]) [congruent to] F(N) [direct sum] [F.sub.N] (N [union] [M.sup.+])

and using Proposition 2

F(M) [congruent to] F(N) [direct sum] [F.sub.N](M) [congruent to]F(N) [direct sum] [F.sub.N](N [union] [M.sup.+]) [direct sum] [F.sub.N](N [union] [M.sup.-])

Since [F.sub.N](N [union] [M.sup.+]) [congruent to] [F.sub.N](N [union] [M.sup.-]) and F(N) [congruent to] F(N) [direct sum] F(N) the result follows.

Theorem 7. Let N be a net in [R.sup.n]. Then F (N) [congruent to] F (N) [direct sum] F (N).

Proof. For n = 1 it is well known  that F (N) [congruent to] F ([N.sup.+]) [congruent to] [[??].sub.1] [congruent to] F (N) [direct sum] F (N).

Inductive step towards n + 1. We may assume that N = [Z.sup.n+1] is the integer grid. Let us use the following notation (our convention is that [Z.sup.+] = {1,2,3,... }, [Z.sup.-] = {-1, -2,... }).

[??] = [Z.sup.n-1] x {0} x {0}, [[??].sub.1] = [Z.sup.n-1] x [Z.sup.+] x {0}, [[??].sub.2] = [Z.sup.n-1] x {0} x [Z.sup.+], [[??].sub.3] = [Z.sup.n-1] x [Z.sup.-] x {0} [M.sup.+] = [Z.sup.n-1] x Z x [Z.sup.+], [M.sub.1] = [Z.sup.n-1] x [Z.sup.+] x [Z.sup.+], [M.sub.2] = [Z.sup.n-1] x [Z.sup.-] x [Z.sup.+]

With this notation, we have the following bi-Lipschitz equivalence

[[??].sub.1] [union] [??] [union] [[??].sub.2] [congruent to] [[??].sub.1] [union] [??] [union] [[??].sub.3].

By inductive assumption this implies

F([??] [union] [[??].sub.1] [union] [[??].sub.2]) [congruent to] F([??] [union] [[??].sub.1] [union] [[??].sub.2]) [direct sum] F([??] [union] [[??].sub.1] [union] [[??].sub.2]). (1)

On the other hand, using Proposition 2 in various settings

F([??] [union] [[??].sub.1] [union] [[??].sub.2]) [congruent to] F([??]) [direct sum] [F.sub.[??]] ([??] [union] [[??].sub.1]) [direct sum] [F.sub.[??]] ([??] [union] [[??].sub.2]),

[mathematical expression not reproducible] (2)

[mathematical expression not reproducible] (3)

Hence, using the inductive assumption F([??] [union] [[??].sub.1] [union] [[??].sub.3]) [congruent to] F([??] [union] [[??].sub.1])

F([??] [union] [[??].sub.1] [union] [[??].sub.2] [union] [[??].sub.3]) [congruent to] F([??] [union] [[??].sub.1]) [direct sum] [F.sub.[??]]([??] [union] [[??].sub.2]) (4)

Comparing (2), (4) and using (1) we obtain

F([??] [union] [[??].sub.1] [union] [[??].sub.2] [union] [[??].sub.3]) [congruent to] F([??] [union] [[??].sub.1] [union] [[??].sub.2]) [congruent to] F([??] [union] [[??].sub.1] [union] [[??].sub.2]) [direct sum] F([??] [union] [[??].sub.1] [union] [[??].sub.2]) (5)

By Lemma 6, in order to complete the inductive step, it suffices to prove that F([M.sup.+]) = F([M.sup.+]) [direct sum] F([M.sup.+]).

Denote R : [R.sup.+] x [R.sup.+] [right arrow] R x [R.sup.+] the mapping R(z) = [z.sup.2]/|z|, where z is the complex number represented as z = x + iy. It is clear that R is bi-Lipschitz. Indeed, if [z.sub.0] = a + ib and [z.sub.1] = x + iy are two complex numbers from the first quadrant with a [less than or equal to] x, then

[mathematical expression not reproducible].

On the other hand, for any [z.sub.0] = a + ib and [z.sub.1] = x + iy from the upper half plane with a [less than or equal to] x we have

[mathematical expression not reproducible],

which we wanted to prove.

The mapping

T : [M.sub.1] [right arrow] [R.sup.n+1], T(u,x,y) = (u, R(x,y))

takes the net [M.sub.1] from the set [R.sup.n-1] x [R.sup.+] x [R.sup.+] in a bi-Lipschitz way to the net T([M.sub.1]) in the set [R.sup.n-1] x R x [R.sup.+]. Hence F([M.sub.1]) [congruent to] F(T([M.sub.1])). Since [M.sup.+] = [M.sub.1] [union] [[??].sub.2] [union] [M.sub.2] is another net in the second set, by Proposition 5 we obtain

F ([M.sub.1]) [congruent to] F ([M.sup.+])

Now thanks to the bi-Lipschitz equivalence [M.sub.1] [congruent to] [M.sub.1] [union] [??] [union] [[??].sub.1] [union] [[??].sub.2],

[mathematical expression not reproducible]

Since [M.sup.+] is bi-Lipschitz equivalent to [M.sup.+] [union] [??] [union] [[??].sub.1] [union] [[??].sub.2] we get

[mathematical expression not reproducible] (6)

Using (5) and the obvious

[mathematical expression not reproducible]

we finally obtain

F([M.sub.1]) [direct sum] F([M.sub.1]) [congruent to] F([M.sup.+]) = F([M.sub.1])

which ends the inductive step and the proof.

Theorem 8. Let X be a Banach space such that X [congruent to] Y [direct sum] X, where Y is an infinite dimensional Banach space with a Schauder basis. Let N be a net in X. Then

[mathematical expression not reproducible].

Proof. We may assume without loss of generality that the norm of the direct sum Y [direct sum] X is in fact equal to the maximum norm Y [[direct sum].sub.[infinity]] X. Using Proposition 5 it suffices to prove the result for just one particular net N. Let [M.sub.k] [subset] [kS.sub.X], k [member of] N be a (1, 2)-net. Then N = [[union].sup.[infinity].sub.k=1] [M.sub.k] is a (1,3)-net in X. Let {[e.sub.k]} be a bi-monotone normalized Schauder basis of Y. Set [mathematical expression not reproducible]. It is clear that

[mathematical expression not reproducible]

We will use Pelczynski's decomposition technique to prove the theorem. Since F(N) is complemented in Z it only remains to prove that F (N) contains a complemented subspace isomorphic to Z. Let

[V.sub.n] = {[ke.sub.n] [direct sum] x : x [member of] [M.sub.k], k [member of] N} [subset] Y [direct sum] X M = [[union].sup.[infinity].sub.n=1][V.sub.n]

The sets [V.sub.n], as subsets of the pointed metric space (Y [direct sum] X, || * ||, 0), satisfy the assumptions of Proposition 2 and so

[mathematical expression not reproducible].

We extend the set M into a (1,3)-net M in Y [direct sum] X. Because F(M) [congruent to] F(N) it suffices to show that F(M) contains a complemented copy of F(M). To this end it is enough to find a Lipschitz retraction R : M [right arrow] M. Denote by [a] the integer part of a [member of] R. First let r : X [right arrow] N be a (non-continuous) retraction such that [||x||] [less than or equal to] ||r(x) || [less than or equal to] ||x||, ||r(x) - x|| [less than or equal to] 4 and ||r(x) || = ||x|| provided ||x|| [member of] N. Let s : Y [right arrow] Y be a (non-continuous) retraction defined for x = [[summation].sup.[infinity].sub.I=1] [x.sub.i][e.sub.i] by

[mathematical expression not reproducible] (7)

It is easy to see that || r(x) - r(y)|| [less than or equal to] 9||x - y||, ||s(x) - s(y)|| [less than or equal to] 6 ||x - y|| provided ||x - y|| [greater than or equal to] 1 (i.e. they are Lipschitz for large distances). Indeed,

||r(x) - r(y) || [less than or equal to] ||r(x) - x|| + ||r(y) - y|| + ||x - y|| [less than or equal to] 8 + ||x - y|| [less than or equal to] 9||x - y||

Assuming 1 [less than or equal to] ||x - y|| [less than or equal to] [lambda], we get |[x.sub.i] - [y.sub.i]| [less than or equal to] [lambda], i [member of] N. Suppose that s(x) = [de.sub.k], s(y) = [te.sub.l]. We claim that d [less than or equal to] 3[lambda]. Indeed, assuming by contradiction that [x.sub.k] [greater than or equal to] d + max {[x.sub.i] : i [not equal to] k} [greater than or equal to] 3[lambda] + max {[x.sub.i] : i [not equal to] k} we obtain that [y.sub.k] [greater than or equal to] [lambda] + max {[y.sub.i] : i [not equal to] k}. Hence k = l and |d - t| [less than or equal to] 2[lambda] + 2. The same argument yields t [less than or equal to] 3[lambda], so finally we obtain ||s(x) - s(y) || [less than or equal to] 6[lambda].

Let R : M [right arrow] M is now defined as

[mathematical expression not reproducible] (8)

We claim that R is a retraction onto M. If y [direct sum] x [member of] M then clearly s(y) = y, r(x) = x, ||s(y)|| = ||r(x)|| and so R(y [direct sum] x) = y [direct sum] x. Next, observe that R(y [direct sum] x) [member of] M holds for every y [direct sum] x [member of] M. Indeed, regardless of the case in the definition of R, we see that the first summand of R(y [direct sum] x) is a non-negative integer multiple of some basis vector [e.sub.n] in Y. In the first (and third) case it is obvious, in the second case it follows as the norm of ||r(x)||/||s(y)|| s(y) is an integer ||r(x)||. The second summand is the result of an application of the retraction r, and its norm equals the norm of the first summand, hence the value of R(y [direct sum] x) indeed lies in M.

Next, we claim that R is Lipschitz. Recall that M is a (1,3)-net in a Banach space, so it suffices to prove that there exists a K > 0 such that ||R([Y.sub.1] [direct sum] x1[x.sub.1]) R( y [direct sum] x)|| [less than or equal to] K whenever ||[y.sub.1] [direct sum] [x.sub.1] - y [direct sum] x|| [less than or equal to] D, for say D = 8. This is well-known and easy to see, as every pair of distinct points p, q G M can be connected by a straight segment of length ||p - q||, and a sequence of [||p - q||] + 1 points on this segment of distance (of consecutive elements) at most one. Each of these points has an approximant from M of distance at most 3, so it clear that there exists a sequence of [||p - q||] + 1 points in M of (consecutive) distance at most D - 1 = 7, "connecting" the points p, q, and the result follows by a simple summation of the increments of R along the mentioned sequence.

Let us start the proof of Lipschitzness of R by partitioning M into three disjoint subsets

[D.sub.1] = {y [direct sum] x : ||x|| > ||s(y)|| [greater than or equal to] 20D}, [D.sub.2] = {y [direct sum] x : ||s(y)|| [greater than or equal to] ||x|| [greater than or equal to] 20D}, [D.sub.3] = {y [direct sum] x : min {||s(y)||, ||x||} < 20D}.

The set [D.sub.1] (resp. [D.sub.2]) corresponds to the case 1 (resp. 2) in the definition of R.

Observe that ||R(y [direct sum] x)|| [less than or equal to] min{||y||, ||x||} so it suffices to prove the Lipschitzness of R on the set [D.sub.1] [union] [D.sub.2]. Moreover, the sets [D.sub.1] and [D.sub.2] have in a sense a common "boundary" (in the intuitive sense, which is not contained in [D.sub.1]) consisting of those elements for which ||x|| = ||s(y)||. It is easy to see that for such elements the first two cases in definition of R may be applied with the same result (although formally we are forced to apply the second case). Suppose now that p [member of] [D.sub.1], q [member of] [D.sub.2]. A similar argument as above using the straight segment connecting p, q (and a short finite sequence from M which approximates this segment) we see that the segment essentially has to "cross the boundary" between [D.sub.1], [D.sub.1], and so the proof of the Lipschitzness of R will follow provided we can do it for each of the sets [D.sub.1], [D.sub.1] separately.

Suppose [y.sub.1] = y + [??], [x.sub.1] = x + [??] are such that ||[??]||, ||[??]|| [less than or equal to] D.

Case 1. We consider first the case [y.sub.1] [direct sum] [x.sub.1], y [direct sum] x [member of] [D.sub.1]. Then

[mathematical expression not reproducible]

Now

[mathematical expression not reproducible]

Similarly, we obtain

||s(y)||/||x|| - ||s(y)|| - 9D/||x|| + D [less than or equal to] 10D/||x||

Hence we obtain

| ||s(y + [??])||/||x + [??]|| - ||s(y)||/||x|| | [less than or equal to] 10D/||x||

The last term is also estimated similarly:

||s(y + [??])||/||x + [??]|| ||[??]|| [less than or equal to] ||s(y)|| + 9D/||x|| - D D [less than or equal to] ||x|| + 9D/||x|| - D D [less than or equal to] 3D

So combining the above computations we get

[mathematical expression not reproducible]

So the mapping [mathematical expression not reproducible] from [D.sub.1] to M takes vectors of distance at most D to vectors of distance at most 15D. It is now clear that R is Lipschitz on [D.sub.1].

Case 2. We consider now [y.sub.1] [??] [x.sub.1], y [??] x [member of] [D.sub.2], and denote z = s(y + [??]) - s(y) (recall that [parallel]z[parallel] [less than or equal to] 9D):

[mathematical expression not reproducible]

Therefore

[mathematical expression not reproducible]

The first term could be rewritten and estimated as follows:

[mathematical expression not reproducible]

The second term we estimate analogously

[mathematical expression not reproducible]

We conclude that R is Lipschitz on the whole domain M. Hence F(M) is isomorphic to a complemented subspace of F{M.) [congruent to] F(N).

A simple situation which fits the above assumptions is when X contains a complemented subspace with a symmetric basis (e.g. [l.sub.p], [c.sub.0] or an Orlicz sequence space). By the standard structural theorems for classical Banach spaces () we obtain.

Corollary 9. Let X be a Banach space isomorphic to any of the (classical) spaces [l.sub.p], [L.sub.p], 1 [less than or equal to] p < [infinity], C(K), or an Orlicz space [h.sub.M], N be a net in X. Then

[mathematical expression not reproducible]

Recall that a metric space M is an absolute Lipschitz retract if, for some K > 0, M is a K-Lipschitz retract of every metric superspace M [subset] N ( p.13). We are going to use the discretized form of this condition. This concept is almost explicit in the work of Kalton , where it would probably be called absolute coarse retract.

Definition 3. Let M be a [delta]-uniformly discrete space, [delta] > 0. We say that M is an absolute uniformly discrete Lipschitz retract if, for some K > 0, the space M is a K-Lipschitz retract of every [delta]-uniformly discrete superspace M [subset] N.

Lemma 10. Let X be Banach space which is an absolute Lipschitz retract, N be a net in X. Then N is absolute uniformly discrete Lipschitz retract. Conversely, if N is absolute uniformly discrete Lipschitz retract and X is a Lipschitz retract of X** then X is an absolute Lipschitz retract.

Proof. The first implication is obvious. To prove the second one, suppose that X [subset] [l.sub.[infinity]] ([GAMMA]) = [UPSILON] is a linear embedding. Since [l.sub.[infinity]] ([GAMMA]) is an injective space, it suffices to prove that there is a Lipschitz retraction from [l.sub.[infinity]] ([GAMMA]) onto X. Since X is a Lipschitz retract of X**, it suffices to follow verbatim the proof of Theorem 1 in . Indeed, consider a net N in X with extension into a net M in [UPSILON]. By assumption, there exists a Lipschitz retraction r : M [right arrow] N. This retraction r can be easily extended to a coarsely continuous retraction R from [UPSILON] onto X (using the terminology of ), which is of course Lipschitz for large distances. It is this condition on R that is used in the proof of Theorem 1 in .

Remark. It is an open problem if the retraction from X (**) to X exists for every separable Banach space (see ).

Important examples of absolute uniformly discrete Lipschitz retract are the nets in C(K) spaces, K metrizable compact,  p.15.

Corollary 11. Let M be a countable absolute uniformly discrete Lipschitz retract which contains a bi-Lipschitz copy of the net N in [c.sub.0]. Then F(M) [congruent to] F(N).

Proof. There is a Lipschitz retraction from M onto N, and on the other hand using Aharoni's theorem ( p. 546) M is bi-Lipschitz embedded into N (and hence also a retract). Thus F(M) is complemented in F(N) and vice versa. To finish, apply Theorem 8 for [c.sub.0] together with the Pelczynski decomposition principle.

To give concrete applications of the above corollary, we obtain the following result. The case of [c.sup.+.sub.0] follows from the Pelant [c.sup.+.sub.0]-version of Aharoni's result .

Theorem 12. Let N be a net in [c.sub.0] and M be a net in any of the following metric spaces: C(K), K infinite metrizable compact, or [c.sup.+.sub.0] (the subset of [c.sub.0] consisting of elements with non-negative coordinates). Then F(M) [congruent to] F(N).

3 Schauder basis

Theorem 13. Let X be a metric space. Suppose there exist a set M [??] X and a sequence of distinct points [{[[mu].sub.n]}.sup.[infinity].sub.n=1] [??] M, together with a sequence of retractions [{[[phi].sub.n]}.sup.[infinity].sub.n=1], [[phi].sub.n] : M [right arrow] M,n [member of] N, which satisfy the following conditions:

(i) [[phi].sub.n] (M) = [M.sub.n] := [[union].sup.n.sub.j=1] {[[mu].sub.j]} for every n [member of] N,

(ii) [mathematical expression not reproducible]

(iii) There exists K > 0 such that [[phi].sub.n] is K-Lipschitz for every n [member of] N,

(iv) [[phi].sub.m][[phi].sub.n] = [[phi].sub.n][[phi].sub.m] = [[phi].sub.n] for every m, n [member of] N,n [less than or equal to] m.

Then the space F(M) has a Schauder basis with the basis constant at most K.

Proof. It is a well-known fact that every Lipschitz mapping L : A [right arrow] B between pointed metric spaces A, B, such that L(0) = 0 extends uniquely to a linear mapping

[??] : F(A) [right arrow] F(B) in a way that that the following diagram commutes:

[mathematical expression not reproducible]

Moreover, the norm of [??] is at most Lip(L). Therefore for every n [member of] N there is a linear mapping [mathematical expression not reproducible] extending [[phi].sub.n] : M [right arrow] M with [parallel][P.sub.n][parallel] [less than or equal to] K. We want to prove that {[P.sub.n]} is a sequence of canonical projections associated with some Schauder basis of F(M), namely that

a) dim [P.sub.n](F(M)) = n - 1 for every n [member of] N,

b) [P.sub.n][P.sub.m] = [P.sub.m][P.sub.n] = [P.sub.m] for all m, n [member of] N, m [less than or equal to] n,

c) [lim.sub.n] [P.sub.n] (x) = x for all x [member of] F(M).

The first condition is easy: as [[phi].sub.n](M) = [M.sub.n] = [{[[mu].sub.i]}.sup.n.sub.i=1] we have [P.sub.n](F(M)) = F([M.sub.n]), which is a (n - 1)-dimensional space. Let us check the commutativity. Note first that for m, n [member of] N the diagram

[mathematical expression not reproducible]

commutes, which means that [mathematical expression not reproducible]. But then from the condition iv follows [P.sub.n][P.sub.m] = [P.sub.m][P.sub.n] = [P.sub.m] for m [less than or equal to] n.

The validity of the limit equation is proved easily. Note that elements of the form [mathematical expression not reproducible], where m [member of] N, [x.sub.i] [member of] [{[[mu].sub.n]}.sup.[infinity].sub.n=1], [[alpha].sub.i] [member of] R for all i [member of] {1, ***, m}, are norm dense in F(M). Indeed, it is a well-known fact that elements [mu] [member of] F(M) of the same form [mathematical expression not reproducible] with [x.sub.i] [member of] M are norm dense in F(M) and the condition ii gives the more general result. By uniform boundedness of the family [{[P.sub.n]}.sup.[infinity].sub.n=1], it suffices to check the limit for elements mentioned above. Thus pick a measure [mathematical expression not reproducible] for all i [member of] {1,..., m}. Find k [member of] N such that {[x.sub.1], ***, [x.sub.m]} [??] [M.sub.k]. Then for all n [greater than or equal to] k we have

[mathematical expression not reproducible]

This was to prove.

Definition 4. Let X be a Banach space with a Schauder basis E = [{[e.sub.i]}.sup.[infinity].sub.i=1]. The set M(E) = {x [member of] X| x = [[summation].sup.[infinity].sub.i=1] [x.sub.i][e.sub.i], [x.sub.i] [member of] Z, i [member of] N} we call the integer-grid to the basis E. If it is clear what basis we are working with, we will denote the set M and speak simply about a grid.

It is not difficult to see that if a basis E is normalized, then the grid M(E) is a [1/2bc(E)]-seParated set, where bc(E) denotes the basis constant of E. For E an unconditional basis we will denote uc( E) the unconditional constant of E. We will now show that for a normalized, unconditional basis E the space F(M) has a Schauder basis.

Lemma 14. Let X be a Banach space with a normalized, unconditional Schauder basis E = [{[e.sub.i]}.sub.i[member of]N] and a grid M(E) = M. Then there exists a sequence of retractions [[phi].sub.n] : M [right arrow] M together with a sequence of distinct points [[mu].sub.n] [member of] M, n [member of] N satisfying the conditions from the Theorem 13 with the constant at most K = uc(E) + 2bc(E).

Proof. Before we define the retractions [{[[phi].sub.n]}.sup.[infinity].sub.n=1] and the points [{[[mu].sub.n]}.sup.[infinity].sub.n=1] rigorously, let us give the reader some geometric idea of how will the retractions look like. We will add points from M so that first the set [C.sup.1.sub.1] = {[x.sub.1] [e.sub.1]| | [x.sub.1]| [less than or equal to] 1} is created, then the set [C.sup.2.sub.1] = {[x.sub.1][e.sub.1] + [x.sub.2][e.sub.2]| |[x.sub.i]| [less than or equal to] 1, i = 1, 2}, then the set [C.sup.2.sub.2] = {[x.sub.1] [e.sub.1] + [x.sub.2] [e.sub.2]||[x.sub.i]| [less than or equal to] 2,i=1,2}, then [C.sup.3.sub.2] = {[[summation].sup.3.sub.i=1] [x.sub.i][e.sub.i]| |[x.sub.i]| [less than or equal to] 2, i = 1,2,3} and so on. Note that coordinates of each [mu] [member of] [C.sup.j.sub.i] are entire numbers.

The retractions will cut coordinates of the argument so that if x = [summation].sup.[infinity].sub.i=1] [x.sub.i][e.sub.i] [member of] M and [{[[mu].sub.i]}.sup.n.sub.i=1] = [M.sub.n] = [[phi].sub.n] (M), n [member of] N, then [[phi].sub.n](x) is obtained by following algorithm: Choose all [[mu].sub.i] [member of] [M.sub.n] minimizing the value |[x.sub.1] - [([[mu].sub.i]).sub.1]|, out of them choose those [mathematical expression not reproducible] minimizing [mathematical expression not reproducible] and so on. Note the process will stop eventually because x = [[summation].sup.k.sub.i=1] [x.sub.i][e.sub.i] for some k [member of] N as x [member of] M and the basis E is normalized. It will be a matter of choosing (ordering) the points [{[[mu].sub.i]}.sup.[infinity].sub.i=1] so that the process ends with only one point [[mu].sub.i] = [[phi].sub.n](x).

We are now going to describe the construction of the sequence [[phi].sub.n] in the following way. We will build the sequence of points [[mu].sub.n] and to each n [member of] N, we associate the sets [[phi].sup.-1.sub.n]([[mu].sub.i]), i [member of] {1, ***,n}. As we want the image [[phi].sub.n](M) = [M.sub.n] = [[union].sup.n.sub.i=1] {[[mu].sub.i]}, the only things needed for the mapping pn to be well-defined is to check U (rn)=1{p (-1) (ji)} = M and [[phi].sup.-1.sub.n]([[mu].sub.i]) [intersection] [[phi].sup.-1.sub.n]([[mu].sub.j]) = [empty set] for i [not equal to] j. For simplicity, we denote the set-valued mapping [[phi].sup.-1.sub.n] = [F.sub.n] and we will define the mappings [[phi].sub.n], n [member of] N through defining [F.sub.n] : [M.sub.n] [right arrow] [2.sup.M]. Note that if for every i [member of] {1,..., n} holds [[mu].sub.i] [member of] [F.sub.n] ([[mu].sub.i]), then the mapping [[phi].sub.n] is a retraction.

In the sequel, by the n-tuple ([a.sub.1],[a.sub.2],...[a.sub.n]), [a.sub.i] [member of] R we will mean the linear combination [[summation].sup.n.sub.i=1] [a.sub.i][e.sub.i] and for a point x [member of] X, x = [[summation].sup.[infinity].sub.i=1] [x.sub.i][e.sub.i] we will always identify x with ([x.sub.1], [x.sub.2], [x.sub.3],...).

Set

[mathematical expression not reproducible]

It is not difficult to see [[phi].sub.1], [[phi].sub.2], [[phi].sub.3] are retractions satisfying the conditions i,iii,iv from the Theorem 13 with Lipschitz constant which equals to uc(E) [less than or equal to] K. Indeed, for [[phi].sub.1] it is clear as its image is only {0}. For [[phi].sub.2], x, y [member of] M and i [member of] N we have

[mathematical expression not reproducible] (9)

and similarily for n = 3, x [member of] M and i [member of] N we have

[mathematical expression not reproducible]

and therefore for x, y [member of] M

[mathematical expression not reproducible] (10)

Due to the unconditionality of E, it is true that for every x [member of] X and z [member of] R, |z| [less than or equal to] [x.sub.1] holds [parallel] (z, [x.sub.2], [x.sub.3], [x.sub.4], ...)[parallel] [less than or equal to] uc(E) [parallel]x[parallel]. But for every i [member of] N the expression in (10) is less or equal to |[x.sub.i] - [y.sub.i]|, which gives us Lipschitz condition on [[phi].sub.n] with constant uc(E).

Moreover, the last retraction [[phi].sub.3] maps M onto the set [C.sup.1.sub.1] [??] M containing all points x [member of] M with x = ([x.sub.1]) and |[x.sub.1]| [less than or equal to] 1. Let us denote [C.sup.d.sub.r] = {x [member of] M| x = ([x.sub.1], [x.sub.2], ..., [x.sub.d]), |[x.sub.i]| [less than or equal to] r, i [less than or equal to] d}. From now on, we will proceed inductively. Suppose we have a sequence of retractions [{[[phi].sub.i]}.sup.m.sub.i=1] together with the points [[mu].sub.i], such that [[phi].sub.m](M) = [C.sup.r.sub.r] and that [{[[phi].sub.i]}.sup.m.sub.i=1] satisfy the conditions i,iii,iv from the Theorem 13. Note that m = [(2r + 1).sup.r].

We proceed by induction which we divide into two steps. First we find points [[mu].sub.m+1],...,[[mu].sub.s] together with retractions [[phi].sub.m+1],...,[[phi].sub.s], where s = [(2r + 1).sup.r|1], such that [M.sub.s] = [C.sup.r|1.sub.r] and such that [{[[phi].sub.i]}.sup.s.sub.i-1] satisfy the conditions i,iii,iv from theorem 13. Then we find points [[mu].sub.s+1],..., and retractions [[phi].sub.s+1],..., [[phi].sub.t] where t = [(2r + 3).sup.r+1], [[phi].sub.t] : M [right arrow] [C.sup.r|1.sub.r|1] which satisfy i,iii,iv. As [[union].sup[infinity].sub.r 1 [C.sup.r.sub.r] = M, the condition ii from theorem 13 is obtained as well, which will conclude the proof.

On the bounded set [C.sup.r.sub.r] we define an ordering by the formula

([x.sub.1], [x.sub.2],..., [x.sub.r]) [??] ([y.sub.1], [y.sub.2],..., [y.sub.r]) [??] ([x.sub.1] > [y.sub.1])[disjunction] [there exists]i [member of]{1,...,r - 1}[for all]j [member of] {1,...,i} : ([x.sub.j] = [y.sub.j]) [and] ([x.sub.i+1] > [y.sub.i+1]). (11)

There exists a bijection w : {1,...,[(2r + 1).sup.r]} [right arrow] [C.sup.r.sub.r], which preserves order.

Let us shorten the notation by introducing indexing functions a, b. If j [member of] {1,...,r} and i [member of] {1,..., [(2r + 1).sup.r]}, let a(j,i) = j[(2r + 1).sup.r] + i and b(j,i) = (r + j)[(2r + 1).sup.r] + i. We set [[mu].sub.a(j,i)] = (w(i), j) = w(i) + j[e.sub.r+1] and [[mu].sub.a(j,i)] = (w(i), - j) = w(i) - j[e.sub.r+1]. Moreover, we formally put [[mu].sub.a(0,i)] = [[mu].sub.b(0,i)] = w(i). Then we define sets

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

It is easy to see that the formulae above define mappings [[phi].sub.a(j,i)] and [[phi].sub.b(j,i)]. Supposed it holds for the mappings [{[[phi].sub.i]}.sup.m.sub.i 1 it is clear that [F.sub.n] ([[mu].sub.p]) [intersection] [F.sub.n] ([[mu].sub.q]) = [??] for p [not equal to] q and all n [member of] {1,..., s}, and that [[mu].sub.n] [member of] [F.sub.n]([[mu].sub.n]) and [[union].sup.n.sub.i=1[[mu].sub.n]([[mu].subi]) = M, which means each mapping [[phi].sub.n] is well-defined and is a retraction onto [M.sub.n].

Let us check the uniform Lipschitz boundedness. Fix n [member of] {m + 1,..., s}. Note first that

[mathematical expression not reproducible]

From this we deduce the Lipschitz boundedness.

If x, y [member of] M, then for i > r + 1 we have |[[phi].sub.n][(x).sub.i] - [[phi].sub.n][(y).sub.i]| = |0 - 0| = 0 [less than or equal to] |[x.sub.i] - [y.sub.i]|.

If i < r + 1 then we distinguish three cases:

a) |[x.sub.i]|, |[y.sub.i]| [less than or equal to] r. Then [[phi].sub.n] [(x).sub.i] = [x.sub.i], [[phi].sub.n] [(y).sub.i] = [y.sub.i] and therefore |[[phi].sub.n][(x).sub.i] - [[phi].sub.n][(y).sub.i]| = |[x.sub.i] - [y.sub.i]|.

b) |[x.sub.i]| [less than or equal to] r, |[y.sub.i]| > r. Then [[phi].sub.n][(x).sub.i] = [x.sub.i] and [[phi].sub.n][(y).sub.i] = r sgn([y.sub.i]). Therefore |[[phi].sub.n][(x).sub.i] - [[phi].sub.n][(y).sub.i]| = |[x.sub.i] - rsgn([y.sub.i])| [less than or equal to] |[x.sub.i] - [y.sub.i]|.

c) |[x.sub.i]|, |[y.sub.i]| > r. Then [[phi].sub.n][(x).sub.i] = r sgn([x.sub.i]), [[phi].sub.n][(y).sub.i] = r sgn([y.sub.i]) and therefore

[mathematical expression not reproducible]

Finally, let i = r + 1. If now [x.sub.i][y.sub.i] < 0, then either 0 [less than or equal to] [[phi].sub.n] [(x).sub.i] [less than or equal to] [x.sub.i] and [y.sub.i] [less than or equal to] [[phi].sub.n][(y).sub.i] [less than or equal to] 0 or vice versa. Both options give |[[phi].sub.n][(x).sub.i] - [[phi].sub.n][(y).sub.i]| [less than or equal to] |[x.sub.i] - [y.sub.i]|, which is what we need.

Let [x.sub.i],[y.sub.i] [greater than or equal to] 0. Suppose n = a(j,k) for eligible j,k. Then [[phi].sub.n][(x).sub.i] = j or [[phi].sub.n] [(x).sub.i] = j - 1 or [[phi].sub.n] [(x).sub.i] = [x.sub.i], which occurs whenever 0 [less than or equal to] [x.sub.i] < j - 1. Of course the same holds for y. From this we have either |[[phi].sub.n][(x).sub.i] - [phi][(y).sub.i] | [less than or equal to] |[x.sub.i] - [y.sub.i]| or |[[phi].sub.n][(x).sub.i] - [phi][(y).sub.i]| [less than or equal to] 1. If n = b(j,k) for some j,k, then [[phi].sub.n][(x).sub.i] = r whenever [x.sub.i] [greater than or equal to] r and [[phi].sub.n] [(x).sub.i] = [x.sub.i] whenever [x.sub.i] < r, the same for y. It is clear that |[[phi].sub.n][(x).sub.i] - [phi][(y).sub.i]| [less than or equal to] |[x.sub.i] - [y.sub.i]|.

Let [x.sub.i], [y.sub.i] [less than or equal to] 0. If n = a(j,k) for some j,k, then |[[phi].sub.n][(x).sub.i] - [phi][(y).sub.i]| = |0 - 0| = 0 [less than or equal to] |[x.sub.i] - [y.sub.i]|. If n = b(j,k) for some j,k, then [[phi].sub.n][(x).sub.i] = - j or [[phi].sub.n][(x).sub.i] = - j + 1 or [[phi].sub.n] [(x).sub.i] = [x.sub.i], which holds whenever 0 [greater than or equal to] [x.sub.i] > -j + 1. Again, we get either |[[phi].sub.n][(x).sub.i] - [phi][(y).sub.i]| [less than or equal to] |[x.sub.i] - [y.sub.i]| or |[[phi].sub.n][(x).sub.i] - [phi][(y).sub.i]| [less than or equal to] 1.

To sum up all cases, if x, y [member of] M, then either [x.sub.r+1] = [y.sub.r+1] or not. In the first case we have

[mathematical expression not reproducible] (12)

as [MU] is a [1/2bc(E)]-separated set, while in the [x.sub.r+1] [not equal to] [y.sub.r+1] case we have

[mathematical expression not reproducible] (13)

Considering both cases we get the mapping [[phi].sub.n] is Lipschitz with constant K = uc(E)+ 2bc(E).

It remains to prove that the mappings [{[[phi].sub.n]}.sup.s.sub.n=1] satisfy the commutativity condition iv, provided the mappings [{[[phi].sub.n]}.sup.m.sub.n=1] do. Note that for any m, n [member of] N, m [less than or equal to] n holds

[F.sub.n]([[mu].sub.n]) [intersection] [F.sub.m]([[mu].sub.m]) [member of] {[??], [F.sub.n]([[mu].sub.n])}. (14)

Out of this fact the commutativity follows easily: Consider i < j [member of] {1,...,s}. First, because [[phi].sub.i] is a retraction onto [M.sub.i] and the same holds for [[phi].sub.j] and [M.sub.i], from [M.sub.i] [??] [M.sub.j] follows [[phi].sub.j][[phi].sub.i] = [[phi].sub.i]. It remains to prove [[phi].sub.i][[phi].sub.j](x) = [[phi].sub.i] (x) for every x [member of] [MU].

Take x [member of] M. There exists a maximal finite sequence of indices 1 = [k.sub.0] < ... < [k.sub.l] [less than or equal to] s such that

[mathematical expression not reproducible].

Clearly if c(i) is the biggest index such that [k.sub.c(i)] [less than or equal to] i, then [mathematical expression not reproducible] for all d, c(i) [less than or equal to] d [less than or equal to] l. This applies analogously for [[phi].sub.j] with c(j). From the fact that both x, [mathematical expression not reproducible] we get simply

[mathematical expression not reproducible]

which finishes the proof of commutativity.

To finish the proof, it remains to show the construction of retractions [[phi].sub.s|1],..., [[phi].sub.t], where t = [(2r + 3).sup.r|1], [[phi].sub.t] : M [right arrow] [C.sup.r+1.sub.r|1] which satisfy i,iii,iv.

For i [member of] N let us define an i-predecessor function [p.sub.i] : M [right arrow] M by

[mathematical expression not reproducible]

Now for every j [member of] {1,..., r + 1} we introduce sets

[mathematical expression not reproducible]

Clearly, [A.sub.j,-1], [A.sub.j,1] [??] [C.sup.r|1.sub.r+1 and |[A.sub.j,-1]| = |[A.sub.j,1]| = [(2r + 1).sup.r+1-j][(2r + 3).sup.j-1]. Moreover,

[mathematical expression not reproducible]

and it is a disjoint union. For each j, choose any bijection [W.sub.j] : {1,..., |A.sub.j,1|} [right arrow] [A.sub.j,1] and fix it. Define [[??].sub.j] : {l,...,|A.sub.j,1|} [right arrow] A.sub.j,-1 by [[??].sub.j](i) = {[w.sub.j][(i).sub.1], [w.sub.j][(i).sub.2], ..., -[w.sub.j][(i).sub.j],..., [w.sub.j][(i).sub.r+1]). For simplicity, for j [member of] {1,..., r + 1}, i [member of] {1,..., |[A.sub.j,1]| put

[mathematical expression not reproducible]

Then we finally set [[mu].sub.[alpha](j,i)] = [w.sub.j](i), [[mu].sub.[beta](j,i)] = [[??].sub.j](i). Now we define mappings [{[F.sub.n]}.sup.t.sub.n s|1 via

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Obviously, the upper equations define mappings [[phi].sub.[alpha](j,i)] and [[phi].sub.[beta](j,i)] for all j [member of] {1,..., r + 1} and i [member of] {1,..., |[A.sub.j,1]|}, hence the mappings [{[[phi].sub.n]}.sup.t.sub.n=s|1] are well-defined and it is an easy check that each such [[phi].sub.n] is a retraction onto the set [M.sub.n].

Note that the sets [{[F.sub.n]([[mu].sub.n])}.sup.t.sub.n=1] still satisfy the condition (14) so the commutativity condition iv from theorem 13 is obtained similarly as it was done for retractions [{[[phi].sub.n]}.sup.s.sub.n=1].

It remains to show the mappings are Lipschitz-bounded. Let us for simplicity denote [[beta].sub.k] = [beta](k - 1, |[A.sub.k-1,1]|) for 1 < k [less than or equal to] r + 1 and [[beta].sub.1] = s, the index of first retraction [mathematical expression not reproducible] such that [mathematical expression not reproducible]. Fix n [member of] {s + 1,...,t}. We will prove that there exists at most one j = j(n) [member of] N such that for all l [member of] N, l [not equal to] j and all x, y [member of] M we have |[[phi].sub.n] [(x).sub.l] - [[phi].sub.n][(y).sub.l]| [less than or equal to] [x.sub.l] - [y.sub.l ]| out of which the Lipschitz boundedness of [[phi].sub.n] follows. If n = [alpha](j, i) for some eligible j, i, then for every x [member of] M holds

[mathematical expression not reproducible]

while if n = [beta](j, i) for some j, i, then for every x [member of] M we have

[mathematical expression not reproducible].

If x,y [member of] M, it is not difficult to see that if |[[phi].sub.n][(x).sub.l] - [[phi].sub.n] [(y).sub.l]| > |[x.sub.l] - [y.sub.l], then l = j and [[phi].sub.n][(x).sub.l] = (r + 1) sgn([x.sub.l]), [[phi].sub.n][(y).sub.l] = r sgn([y.sub.l]) or vice versa and [x.sub.l][y.sub.l] > 0. Particularly |[[phi].sub.n] [(x).sub.l] - [[phi].sub.n] [(y).sub.l]| = 1 and |[x.sub.l] - [y.sub.l]| = 0. For all other l, i.e. l [not equal to] j, l [member of] N holds |[[phi].sub.n] [(x).sub.l] - [[phi].sub.n][(y).sub.l]| [less than or equal to] [x.sub.l] - [y.sub.l]|, which is what we need.

Therefore we get by computation similar to those done in (12) and (13) that [[phi].sub.n] is a Lipschitz mapping with constant K = uc(E) + 2bc(E), which concludes the induction.

As [[union].sup.[infinity].sub.r=1 [C.sup.r.sub] = M the condition ii from theorem 13 is also satisfied and hence our proof is finished.

Remark. In (11) it was not necessary for our construction to choose exactly this order. Infact, any bijection w : {1,..., [(2r + 1).sup.r]} [right arrow] [C.sup.r.sub.r] would suit our purpose. We chose the order (11) for simplicity. In this case we have [[mu].sub.a(j-1,i)] = [p.sub.j]([[mu].sub.a(j,i)]) and [[mu].sub.b(j-1,i)] = [p.sub.j]([[mu].sub.b(j,i)]) for [p.sub.j] the j-predecessor function and i [member of] {1,..., [(2r + 1).sup.r]}, j [member of] {1,...,r}.

Corollary 15. If E = [{[e.sub.i]}.sup.[infinity].sub.i=1] denotes the canonical basis in [c.sub.0] and M = M(E) [??] [c.sub.0] the integer grid, then the Free-space F(M) has a monotone Schauder basis.

Proof. applying the construction of the retractions from the lemma 14 to ([c.sub.0], E), we get Lipschitz constant K = 1, (see estimates (12) and (13)). Therefore, F(M) has a monotone Schauder basis.

Corollary 16. Let N [??] [c.sub.0] be a net. Then the Free-space F(N) has a Schauder basis.

Proof. If we use the notation from previous corollary, M is a (1,1)-net in [c.sub.0]. But as all nets in an infinite-dimensional space are Lipschitz equivalent (, p.239, Proposition 10.22), N is Lipschitz equivalent to the grid M and therefore F(N) is isomorphic to F(M), which concludes the proof.

Corollary 17. Let N be a net in any of the following metric spaces: C(K), K metrizable compact, or [c.sup.+.sub.0] (the subset of [c.sub.0] consisting of elements with non-negative coordinates). Then F(N) has a Schauder basis.

Proof. Follows immediately from Theorem 12.

Corollary 18. Let N [??] [R.sup.n] be a net. Then F(N) has a Schauder basis.

Proof. It follows from the proof of lemma 14 that F([Z.sup.n]) has a Schauder basis and F([Z.sub.n]) [congruent to] F(N) by Proposition 5, which gives the result.

Acknowledgment. The authors would like to thank the referee for his careful reading of the original manuscript, and for several useful comments which lead to the improvement in the presentation of the paper. We would also like to thank the editor for his patience.

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Mathematical Institute, Czech Academy of Science

Zitna 25, 115 67 Praha 1, Czech Republic and

Department of Mathematics

Faculty of Electrical Engineering

Czech Technical University in Prague

Zikova 4, 160 00, Prague

email: hajek@math.cas.cz

Department of Mathematics

Faculty of Electrical Engineering

Czech Technical University in Prague

Zikova 4, 160 00, Prague

email: novotny@math.feld.cvut.cz

* The work was supported in part by GACR 16-073785, RVO: 67985840 and by grant SGS15/194/OHK3/3T/13 of CTU in Prague.

Received by the editors in April 2016 - In revised form in November 2016.

Communicated by G. Godefroy.

2010 Mathematics Subject Classification : 46B03, 46B10.