On certain subspaces of (cid:2) p for 0 < p ≤ 1 and their applications to conditional quasi-greedy bases in p -Banach spaces

We construct for each 0 < p ≤ 1 an inﬁnite collection of subspaces of (cid:2) p that extend the example of Lindenstrauss (Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 , 539–542, 1964) of a subspace of (cid:2) 1 with no unconditional basis. The structure of this new class of p -Banach spaces is analyzed and some applications to the general theory of L p -spaces for 0 < p < 1 are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in p -Banach spaces for p < 1. Among the topics we consider are the existence of inﬁnitely many conditional quasi-greedy bases in the spaces (cid:2) p for p ≤ 1 and the careful examination of the conditionality constants of the “natural basis” of these spaces.


Introduction
The subject of finding estimates for the rate of approximation of a function by means of essentially nonlinear algorithms with respect to biorthogonal systems and, in particular, the greedy approximation algorithm using bases, has attracted much attention for the last 20 years, on the one hand from researchers interested in the applied nature of non-linear approximation and, on the other hand from researchers with a more classical Banach space theory background. Although the basic idea behind the concept of a greedy basis had been around for some time, the formal development of a theory of greedy bases was initiated in 1999 by Konyagin and Temlyakov in the important paper [20]. Subsequently, the theory of greedy bases and its derivates developed very fast as many fundamental results were discovered, and new ramifications branched out. As a result, this is an area with a fruitful interplay between abstract methods from classical Banach space theory and other, more concrete techniques, from approximation theory. The reader interested in approximation theory and/or numerical algorithms may consult the paper [29] or the book [30].
In this article we will concentrate on the functional analytic aspects of this theory, where, rather unexpectedly, the theory of greedy bases has links to old classical results and also to some open problems. See [6,Chapter 10] for an introductory approach to the subject from this angle.
To set the mood let us start by recalling the main concepts we will require from approximation theory in the general setting of quasi-Banach spaces. Let Y be a quasi-Banach space and assume that B = ( y n ) ∞ n=1 is a semi-normalized fundamental Mbounded Markushevich basis in Y, that is, B generates the whole space Y and there is a sequence ( y * n ) ∞ n=1 in the dual space Y * such that ( y n , y * n ) ∞ n=1 is a biorthogonal system with inf n y n > 0 and sup n y * n < ∞. From now on we will refer to any such B simply as a basis. A basic sequence will be a sequence in Y which is a basis of its closed linear span. Note that semi-normalized Schauder bases are a particular case of bases. Given A ⊆ N finite, S A = S A [B, Y] : Y → Y will denote the coordinate projection on A with respect to the basis B, For f ∈ Y and m ∈ N we define where A ⊆ N is a set of cardinality m such that | y * n ( f )| ≥ | y * k ( f )| whenever n ∈ A and k / ∈ A. The set A depends on f and may not be unique; if this happens we take any such set. Thus, the operator G m is well-defined, but is not linear nor continuous. The biorthogonal system ( y n , y * n ) ∞ n=1 (or the basis B) is said to be quasi-greedy provided that there is a constant C ≥ 1 so that for every f ∈ Y and for every m ∈ N we have Equivalently, by [32,Theorem 1], ( y n , y * n ) ∞ n=1 is a quasi-greedy system if Of course, unconditional bases are quasi-greedy, but the converse does not hold in general. Konyagin and Telmyakov provided in [20] the first examples of conditional (i.e., not unconditional) quasi-greedy bases. Subsequently, Wojtaszczyk proved in [32] that, for 1 < p < ∞, the space p has a conditional quasi-greedy basis, and Dilworth and Mitra constructed in [12] a conditional quasi-greedy basis of 1 . These papers were the forerunners of an industry devoted to studying the existence of conditional quasi-greedy bases in Banach spaces. The reader will find in the articles [2,3,5,7,9,11,13,15,16,24,32] some of the main achievements in this direction of research. It is important to point out here that not all Banach spaces have a quasi-greedy basis. Indeed, this is the case, for instance, with C([0, 1]) since, by a result of Dilworth et al. [9] the only L ∞ -space with a quasi-greedy basis is c 0 . Banach space old-timers will have made the connection with a classical result of Lindenstrauss and Pełczyński [22] stating that the canonical c 0 -basis is, up to equivalence, the only unconditional basis of an L ∞ -space. From the general point of view of approximation theory, and more specifically the practical implementation of the greedy algorithm for general biorthogonal systems, it is very natural to ask about the existence of conditional quasi-greedy bases in the context of nonlocally convex quasi-Banach spaces. Since L p ([0, 1]) for 0 < p < 1 has trivial dual (making it therefore impossible for L p to have a basis), the first nonlocally convex spaces that come to mind as objects of study for having conditional quasigreedy bases are the spaces p for 0 < p < 1 (see [1,Problem 12.8]). However, the tools that have been developed for building conditional quasi-greedy bases in Banach spaces break down when local convexity is lifted. For instance, the Dilworth-Kalton-Kutzarova method, DKK-method for short, for constructing conditional quasi-greedy bases in a Banach space X ( [9], cf. [2]) relies on the existence of a complemented subspace S of X with a symmetric basis. A careful inspection of the method reveals that the boundedness of the averaging projection with respect to the symmetric basis of S is a key ingredient in the recipe, hence it stops working when S is not locally convex. Since p for p < 1 is prime [28], it is hopeless to try to build a quasi-greedy basis in p by means of the DKK-method.
These initial drawbacks in making headway create a breeding ground for guesswork. Since quasi-greedy bases in quasi-Banach spaces are not too far from being unconditional (they are unconditional for constant coefficients, see [1, Theorems 3.8 and 3.10]) and the standard unit vector system is the unique, up to equivalence, normalized unconditional basis of p [18], one could be tempted to speculate that it will be the unique quasi-greedy basis in p , which would disprove the existence of conditional quasi-greedy bases in the space. In this paper we refute this conjecture and show that indeed such bases of p for p < 1 exist. In fact, the conditional bases we find belong to the more demanding class of almost greedy bases.
The existence of a conditional quasi-greedy basis of p shows in particular that p does not have a unique quasi-greedy basis, so we discuss the question of how many mutually non-equivalent quasi-greedy basis there are in p , 0 < p < 1.
Our construction of conditional almost greedy bases in p for 0 < p < 1, is inspired and at the same time extends the example of Dilworth and Mitra from [12] of a conditional almost greedy basis in 1 . Their example was derived in turn from the basic sequence constructed by Lindenstrauss [21] of a monotone, conditional, basic sequence in 1 whose closed linear span is a L 1 -space which is not isomorphic to 1 and therefore has no unconditional basis. Adapting this script to our context, in Sect. 3 we manufacture for each 0 < p < 1 and each sequence of integers δ = (d n ) ∞ n=1 contained in the interval [2, ∞), a L p -space denoted l p (δ) which is not isomorphic to p ; in particular, l p (δ) does not have an unconditional basis. The spaces l p (δ) do have, however, a Schauder basis X p (δ) whose features are studied in Sect. 4. We prove that for each 0 < p ≤ 1 and each sequence δ, the basis X p (δ) is quasi-greedy and superdemocratic, hence almost greedy. As a by-product of our work we identify the q-Banach envelope of the spaces l p (δ), 0 < p < q ≤ 1, as being q .
To quantify the conditionality of a quasi-greedy basis B in a quasi-Banach space Y, in Sect. 5 we study the growth of the constants It follows from a result of Dilworth et al. [9,Lemma 8.2] that quasi-greedy bases in Banach spaces cannot be "too conditional" in the sense that they satisfy the estimate (1.1) Moreover, there are examples of quasi-greedy bases in certain Banach spaces for which the logarithmic growth is actually attained (see [14,Sect. 6]). More recently, it was noticed in [4] that there is a entire class of spaces, namely super-reflexive Banach spaces, on which (1.1) can be improved to for some 0 < < 1. Taking into consideration the role played by the convexity of the space in the proof of inequality (1.1), it is not surprising that the conditionality constants of quasi-greedy bases in general p-Banach spaces satisfy the estimate We will see this in Corollary 5.1 after we set in motion the machinery specific to nonlocally convex spaces to prove it. Next, the optimality of (1.2) within p-Banach spaces is established by proving that the reverse inequality, (log m) 1/ p = O(k m ), is attained in p for some conditional quasi-greedy basis. These discussions naturally lead to attempts to construct a conditional quasi-greedy basis B with prescribed growth of k m . In Sect. 6 we develop new techniques to produce such bases in p and l p (δ) (see Theorem 6.2). Since, roughly speaking, the growth of the sequence (k m ) ∞ m=1 is stable under (permutative) equivalence of bases, i.e., our result yields the existence of uncountably many (permutatively) non-equivalent quasi-greedy basis in p and l p (δ) (Corollary 6.2). The novelty in our approach has to be seen also in that our techniques are valid for the limit case p = 1. This nicely complements the main result from [8], where Dilworth et al. showed that if 1 ≤ p < ∞ then p has a continuum of permutatively non-equivalent almost greedy bases.
We close in Sect. 7 with a qualitative and quantitative study of dual Lindenstrauss bases that gives continuity to the results on the subject by Berná et al. [7].

Notation and terminology
Throughout this paper we use standard facts and notation from Banach spaces and approximation theory (see e.g. [6]). The reader will find the required specialized background and notation on greedy-like bases in quasi-Banach spaces in the recent article [1]; however a few remarks are in order.
We write F for the real or complex scalar field. As is customary, we put δ k,n = 1 if k = n and δ k,n = 0 otherwise. The unit vector system of F N will be denoted by and supp(x) will be the set {n ∈ N : x(n) = 0}. We will denote by ·, · the natural pairing in F N × F N , that is, we put (a n ) ∞ n=1 , (b n ) ∞ n=1 = ∞ n=1 a n b n whenever the series ∞ n=1 a n b n converges.
If A ⊆ N is finite and ε = (ε n ) n∈A are signs, we put If ε n = 1 for all n ∈ A we denote 1 ε,A simply by 1 A . The same symbol 1 A will be used as well to denote the indicator function of a measurable set A ⊆ [0, 1]. If (B n ) ∞ n=1 are Markushevich bases in quasi-Banach spaces (Y n ) ∞ n=1 and 0 < q < ∞, ∞ n=1 B n q denotes the obvious Markushevich basis in the quasi-Banach space ∞ n=1 Y n q . If B n = B for every n ∈ N, we put q (B) = ∞ n=1 B n q . We say that two Markushevich bases We say that the Markushevich bases B 1 and B 2 are permutatively equivalent, and we write B 1 ∼ B 2 , if there is a bijection π : N → N such that ( y π( j) ) ∞ j=1 and B 2 are equivalent.
The quasi-norm of a linear operator T between two quasi-Banach spaces X and Y will be denoted by T X→Y . If X and Y are clear from context, we simply write (1.3) Other more specific notation will be introduced when needed.
2 Preliminaries on L p -spaces for 0 < p < 1 L p -spaces (1 ≤ p ≤ ∞) were introduced in [22] by Lindenstrauss and Pełczyński as those Banach spaces whose local structure resembles that of the spaces p . Thus a Banach space X is an L p -space if there is a constant λ such that for every finite dimensional subspace V of X there is a finite dimensional subspace W containing V and a linear isomorphism T : W → n p with T T −1 ≤ λ. For 0 < p < 1, the theory of L p -spaces was developed in [19] by Kalton, who gave an alternative definition more suitable for p-Banach spaces based on the notion of local complementability. Kalton defined a closed subspace Y of a quasi-Banach space X as being locally complemented in X if there is a constant λ such that for every finite-dimensional subspace V of X and every ε > 0 there is a linear operator T : V → Y with T ≤ λ and T | V∩Y − Id V∩Y ≤ ε. Then he went on and defined a quasi Banach space to be an L p -space for 0 < p < 1 if it is isomorphic to a locally complemented subspace of L p (μ) for some measure μ. Since it is unclear whether L p ([0, 1]) for 0 < p < 1 fulfils the condition introduced by Lindenstrauss and Pełczyński, one of the advantages of working with this definition versus adopting the one from the case p ≥ 1 is that it guarantees that the spaces L p ([0, 1]) for p < 1 are L p -spaces.
Let us briefly summarize the relation between the classical definition of L p -spaces by Lindenstrauss and Pełczyński and that of Kalton's depending on p. For p = 1 both definitions are equivalent. For 1 < p < ∞ the difference is that, with Kalton's definition, Hilbert spaces turn out to be L p -spaces. For 0 < p < 1 its is unknown whether Kalton's definition implies the classical one (see the introductory paragraph of [19,Section 6]). Still, we infer from [19, Theorem 6.1] that if a p-Banach space Y (0 < p < 1) possesses an increasing net (V i ) i∈I of finite-dimensional subspaces such that ∪ i∈I V i = Y and V i is uniformly isomorphic to dim(V i ) p , then Y is a L p -space. Since Kalton's paper there has been little effort at a systematic treatment of L pspaces for 0 < p < 1. It is the authors' opinion, however, that these spaces are of interest and therefore deserve such a treatment. The reasons for bringing up L p -spaces for 0 < p < 1 here are twofold. Firstly, they provide a very natural general framework for the new p-Banach spaces that we will introduce below. Secondly, we will extend to p < 1 the classical Lindenstrauss-Pełczyński result on unconditional bases in L 1spaces [22,Theorem 6.1] asserting that the only L 1 -space with an unconditional basis is 1 .
The proof of Theorem 2.1 below relies on the concept of pseudo-dual spaces. Following [19], a quasi-Banach space Y is said to be a pseudo-dual space if there is a Hausdorff vector topology on Y for which the unit ball is relatively compact. By the Banach-Alaoglu theorem, every dual space is a pseudo-dual.

Remark 2.1
For every 0 < p < ∞, p is a separable pseudo-dual space. In fact, every quasi-Banach space Y with a boundedly complete basis B = ( y n ) ∞ n=1 is a pseudo-dual space. To see this, without loss of generality we can assume that B is monotone. Then B Y is compact with respect to the topology of coordinate-wise convergence.
Proof Assume by contradiction that B is an unconditional basis of Y and that Y is not isomorphic to p . Then, by [19,Theorem 6.4], Y is isomorphic to a locally complemented subspace of p which cannot be complemented in p because p is prime [28]. By [19,Theorem 4.4], Y is not a pseudo-dual space, and so by Remark 2.1, the basis B is not boundedly complete. We deduce the existence of an element f in Y and of pairwise disjoint sets (A n ) ∞ n=1 such that By unconditionality it follows that (S A n [B, Y]( f )) ∞ n=1 is equivalent to the canonical basis of c 0 and so c 0 would be isomorphic to a subspace of p , which is an absurdity by Stiles' structural results on p from [28].

A new family of subspaces of p , 0 < p ≤ 1.
In 1964, Lindenstrauss [21] proved the existence of a subspace of 1 , namely ker(Q), where Q is any bounded linear map from 1 onto L 1 ([0, 1]), which is not isomorphic to a dual space. This space provided the first example of a Banach space with a basis but with no unconditional basis, despite being a subspace of a space having an unconditional basis. This section is devoted to generalizing Lindenstrauss example to the range 0 < p < 1. Moreover we will rig our construction in such a way that we produce at once, for every p < 1, an infinite collection of p-spaces fulfilling the desired properties.
The natural way to construct the quotient Q p from p onto To control the properties of Q p , and in particular to be able to handle ker Q p , we must be more specific. We start our construction with a sequence of integers The map thus constructed starting with δ will be denoted by Q p (δ).
Before we get started, some terminology is in order. Given an increasing map In fact, ρ can be defined by We will refer to ρ as to the left inverse of σ . Note also that ρ Given n ∈ N, and a map σ : A ⊆ N → N, we will denote by σ (n) the nth iteration of σ , and we will use the convention σ (0) = Id N . Note that the domain of σ (n) decreases as n increases. Now we start our construction.
and for A ⊆ N put is increasing as well. We will also consider the left inverse of σ , and the left inverse of Λ, We have Λ(0) = 1 and Λ(1) = 2. Therefore, Γ (1) = 0 and Γ (2) = 1. Let us define a partition (J n ) ∞ n=0 of N by thus in the first two steps we defined Λ(2) − 1 functions.
Let us explain the general recursive process. Assume that h j , I j and λ j with λ Let us summarize some properties of this construction and of Q p (δ) that will be used below and that can be directly deduced from the definition.

Fact 3.9 From Fact 3.8 we easily see that for a linear combination x
for any scalar λ.

Fact 3.11 In particular, Fact 3.5 yields that
where, by convention, we put 0 j=1 λ j = 0 and 0 j=1 λ j = 1 for every family (λ j ). Let us next record some elementary properties of ( (3.6) Fact 3. 12 Since σ (1) = 2 we have that J 1,n = J n for every n ∈ N and so (J 1,n ) ∞ n=0 is a partition of N. For a general k ∈ N, since k + 1 ≤ σ (k) we have J k,n < J k,n+1 for every n ≥ 0. In particular, (J k,n ) ∞ n=0 are pairwise disjoint integer intervals. Fact 3.14 x * j ( j) = 1 for every j ∈ N. 18 We infer from Fact 3.16 that, for every j ∈ N, Fact 3.21 Also from Fact 3.19, if j ∈ J k,n then |x * j (k)| ≤ 2 −n/ p . For 0 < p ≤ 1 and δ a sequence of integers in [2, ∞), throughout this paper we will use the following notation where x k is as in (3.4) and x * k is as in (3.5). We will also consider finite-dimensional spaces and finite sequences associated to and We will denote by l * p = l * p (δ) the dual space of l p (δ).

Remark 3.1
The only precedent in the literature for the spaces l p (δ) and the sequences X p (δ) is the case when p = 1 and d n = 2 for n ∈ N. The resulting space for those values is indeed the Lindenstrauss space from [21] to which we referred at the beginning of the section. For other sources alluding to this relevant example, see e.g.
By Facts 3.6, 3.8, 3.16 and 3.14, it suffices to consider the case when k < j.
Proof We proceed by induction on N . If N = j, put x = x j . Assume that the vector x fulfils the desired conditions for some N ≥ j. Then, by Fact 3.11, y = x − S N +1 (x) x N +1 satisfies the condition for N + 1.
Proof It is straightforward from Lemma 3.1.
for every C > 1. By the Open Mapping theorem Q p is onto, hence a quotient map. In fact, the map from p / Ker(Q p ) onto L p induced by Q p is an isometry.
By Fact 3.4, l p = l p (δ) ⊆ Ker(Q p ). Let us prove the reverse inclusion. Let x ∈ p with Q p (x) = 0 and fix ε > 0. There is n ∈ N such that Taking into account that  Proof There is a subsequence (x k n ) ∞ n=1 of X p (δ) which is a block basic sequence with respect to the unit vector system of p . Indeed, it suffices to choose k n = Λ(n) for every n ∈ N. Let us define maps J , P : p → p by where A k = supp(x k ) By Fact 3.5, J is an isometry. Combining Fact 3.15 with the elementary inequality we have for all x ∈ p . We observe that J (e j ) = x n j and P(x n j ) = e j for all j ∈ N. Hence, The following theorem gathers some structural properties of l p (δ)-spaces.
Since L p is not a complemented subspace of p (in fact, L p is not a subspace of p ) we reach a contradiction, thus (c) holds. Since p is an L p -space as well, (d) holds. (e) follows from (b) and [19,Theorem 4.4]. We deduce (f) from (e) and Remark 2.

Remark 3.2
It is not too difficult to come up with examples of non-locally convex spaces with a Schauder basis but without an unconditional basis. Indeed, the space L 1 p , 0 < p < 1, for instance, verifies both conditions. This can be deduced simply by noticing that L 1 does not embed in any quasi-Banach space with an unconditional basis. Indeed, the proof for Banach spaces (see, e.g., [6, Theorem 6.3.3]) remains valid for quasi-Banach spaces. However, one can argue that adding a locally convex component to a nonlocally convex space is cheating. Thus, the question gains interest if we only accept examples within the class of hereditably non-locally convex spaces. Recall that a quasi-Banach space Y is hereditably non-locally convex if every infinite-dimensional subspace of Y is non-locally convex, and that a quasi-Banach space Y is said to be W-saturated if every infinite-dimensional subspace of Y contains a further subspace isomorphic to W. Since p is p -saturated (see [28]), every L p -space with the bounded approximation property, (BAP) for short, is also p -saturated by [19,Theorem 6.4], hence hereditably non-locally convex. Therefore any L p -space with (BAP) which is not isomorphic to p (for instance, the space l p for p < 1) is an example of a hereditably non-locally convex quasi-Banach space with a Schauder basis but without an unconditional basis. To the best of our knowledge, these are the first-known examples of quasi-Banach spaces with these properties.

Bases in the spaces l p (ı)
This section focusses on the sequences X p (δ) constructed in the previous section for p ∈ (0, 1] and for any integers δ = (d n ) ∞ n=1 in the interval [2, ∞). Thanks to Theorem 3.23 we know that the space l p (δ) has a Schauder basis. As a matter of fact, the sequence X p (δ) is a Schauder basis of l p (δ) as we will next prove. Thus, we will rightfully say that X p (δ) is the Lindenstrauss p-basis of l p (δ).
Proof In order to prove that X p (δ) is monotone if suffices to see that x ≤ x +a x n whenever n ∈ N, a ∈ F, and x ∈ l (n−1) p (δ). Let y = x + a x n . From Fact 3.5 and Fact 3.6 we have x( j) = y( j) unless j = n, in which case y( j) = x( j) = a, or j ∈ [σ (n), σ (n + 1)), in which case x( j) = 0 and y( j) = ad −1/ p n . Therefore which combined with inequality (3.7) yields x ≤ y . Now, to finish the proof we need only show that x ≤ x + a x n whenever supp(x) ⊆ [n + 1, ∞). But if y is as before, a similar argument gives via the natural pairing, is the dual basis of X p . Note that Proposition 4.1 yields, in particular, which combined with Fact 3.6, gives Also thanks to Theorem 3.23 we know that the spaces l p are L p -spaces for 0 < p ≤ 1. In hindsight this can be deduced from Proposition 4.2. 0 < p ≤ 1 and δ be a sequence of integers in [2, ∞). For every n ∈ N, the Banach-Mazur distance from l (n) p (δ) to n p is not larger than 2 1/ p .

Proposition 4.2 Let
Proof For each n ∈ N we will recursively construct vectors (x k,n ) n k=1 in l (n) p such that, if r k,n = x k,n − e k , then r k,n p = 1 and supp(r k,n ) ⊆ [n + 1, σ (n + 1)) for k = 1, …, n.
For n = 1 put x 1,1 = x 1 . Let n ∈ N and asume that (x k,n ) n k=1 has been constructed. We set x k,n+1 = x k,n − x k,n (n + 1)x n+1 and x n+1,n+1 = x n+1 . By Fact 3.11, (x k,n+1 ) n+1 k=1 fulfills the desired properties. We easily infer that the vectors (x k,n ) n k=1 satisfy Our construction of conditional almost greedy bases in p will rely on the following isomorphism.

Corollary 4.1 Let 0 < p ≤ 1 and δ be a sequence of integers in [2, ∞). Then for any sequence of positive integers
n k p p , which, in turn, is isometric to p .

Quasi-greediness of Lindenstrauss p-bases, 0 < p < 1.
Our aim in this section is to extend to Lindenstrauss p-bases the main result from [12], where it is proved that the Lindenstrauss basis of the space l 1 (δ), for δ the constant sequence d n = 2, in our notation, is conditional and quasi-greedy. We use G m = G m [ p, δ](x) for the mth greedy projection of x ∈ l p (δ) with respect to the basis X p (δ).

Theorem 4.1 For any 0 < p ≤ 1 and any sequence δ = (d n ) ∞ n=1 of integers in [2, ∞), the Lindenstrauss p-basis X p (δ) is a quasi-greedy basis of l p (δ). Quantitatively, for
x ∈ l p (δ) and m ∈ N, For p = 1, we get the same estimate as in [12]. Note that when p goes to zero the estimate grows as 2 1/ p . Before we tackle the proof of this result, we prove a couple of auxiliary lemmas.

be a sequence of integers in [2, ∞). For every A ⊆ N finite and every x ∈ [x k : k ∈ A] we have x ≤ 2 1/ p S A (x) and x ≤ 2 1/ p S Σ(A) (x) .
Proof Set x = k∈A a k x k . Let y = k∈A a k e k and z = k∈A a k r k , so that x = y + z. Since y = z < ∞, y is supported on A, and z is supported on Σ(A), Lemma 4.1 yields the desired result.

Proof of Theorem 4.1
For k ∈ N put a k = x * k , x . Let A be the mth greedy set of x, so that y := G m (x) = k∈A a k x k and, if B = N \ A, Let z = x − y = k∈B a k x k . By Lemma 4.2, it suffices to prove that and that We split this into three cases: , then z( j) = 0 and, hence, , then y( j) = 0 and, hence, This implies (4.1) and (4.2).

Democracy properties of Lindenstrauss p-bases.
The concepts of democratic and super-democratic bases are by now fairly standard in greedy approximation theory. They have a verbatim translation into the setting of quasi-Banach spaces (see [1, §4])). To quantify the democracy of a basis B in a quasi-Banach space Y, we consider the lower and upper democracy functions of B, given by

The basis B is then super-democratic if and only if
, and it is obvious that if Y is a p-Banach space and B is semi-normalized then Φ u,s m [B, Y] m 1/ p for every m ∈ N. As we next show, in the case when Y = p a reverse inequality holds.

be a sequence of finite-dimensional quasi-Banach spaces, and let
we will refer to f j as the jth coordinate of f . Combining the compactness of each B Y j with Cantor's diagonal technique yields φ : N → N increasing such that ( y φ(k) ) ∞ k=1 converges coordinate-wise. Then ( y φ(2k−1) − y φ(2k) ) ∞ k=1 is a null sequence. A gliding-hump argument yields an increasing sequence ψ : N → N such that B = ( y ψ(2k−1) − y ψ(2k) ) ∞ k=1 is equivalent to a disjointly supported sequence (with respect to the coordinates). Therefore, B is equivalent to the unit vector system of p . By [1, Equation (7.1)], for m ∈ N we have [2, ∞). Then the Lindenstrauss p-basis X p = X p (δ) is a super-democratic basis of l p = l p (δ). Quantitatively, for m ∈ N:

be a sequence of integers in
Proof Let A be a finite subset of N and let γ be a sequence of signs. Since d k ≥ 2 for every k ∈ N, Fact 3.9 yields and so, This establishes (i). Inequality (ii) is clear.

Remark 4.2
It is known that all quasi-greedy bases in 1 and 2 are democratic (see [13,Theorem 4.2] and [32,Theorem 3], respectively) and that if p ∈ (1, 2) ∪ (2, ∞) there are quasi-greedy (even, unconditional) bases of p that are not democratic (see [25]). We emphasize that the techniques developed to settle the question for 1 do not transfer to p for 0 < p < 1, and that all the known examples of quasi-greedy bases in p are democratic. Thus, the following question seems to be open:

Question 4.1
Is every quasi-greedy basis in p for 0 < p < 1 democratic?

Banach envelopes of Lindenstrauss l p (ı) spaces and bases.
When dealing with a quasi-Banach space Y it is often convenient to know what the "smallest" Banach space containing Y is (if there is any), or even what the smallest q-Banach space containing Y is. To be precise, the q-Banach envelope of Y, is a linear contraction, satisfying the following universal property: for every q-Banach space X and every bounded linear map T : n=1 is a basis of Y (q) called the q-envelope basis of B. If X is a q-Banach space and S : Y → X is a bounded linear map such that S (q) : Y (q) → X is an isomorphism, we say that X is isomorphic to the q-Banach envelope of Y via S. We refer the reader to [1, Section 9] for related properties involving these concepts.
Our first result in this section exemplifies how tools from (nonlinear) greedy approximation theory can be efficiently used to deduce functional analytic (linear) properties of bases. Indeed, Proposition 4.4 in combination with [1, Proposition 9.12] immediately yield the following result. Proposition 4.5 Given 0 < p < q ≤ 1, the q-Banach envelope of the basis X p (δ) is equivalent to the canonical basis of q . In other words, the q-Banach envelope of l p (δ) is isomorphic to q via the coefficient transform F : l p (δ) → F N with respect to the basis X p (δ).
Going further, in this section we will show the following.
n=1 be a sequence of integers in [2, ∞). Then the q-Banach envelope of l p (δ) is isomorphic to q via the inclusion map. In particular, l p (δ) is dense in q .
To tackle the proof of this result, we need a couple of lemmas.
n=1 be a sequence of integers in [2, ∞). Then, for every k ∈ N, and every J k,n as in (3.6), Proof (a) is a straightforward consequence of Fact 3.21. We will prove (b) by induction on n. If n = 0, J k,n = {k} and, then, the result follows from Fact 3.14. Assume that n ∈ N and that the result holds for n − 1. Since {σ (J i,n−1 ) : i ∈ [σ (k), σ (k + 1))} is a partition of J k,n , applying Fact 3.20 we obtain

Proof of Theorem 4.2 Define
Combining inequality (3.7), Fubini's theorem, and Lemma 4.4 yields for all x ∈ F N and some constant C. That is, T : q → q is a bounded linear operator that extends the coefficient transform F : l p (δ) → F N with respect to the basis X p (δ) of l p (δ). Therefore, by Proposition 4.5, there is S : q → q such that S • F is the inclusion map J of l p (δ) into q . Pictorially, the diagram q S q T l p (δ) F J is commutative. We infer that T • S = Id q . Since T is one-to-one by Fact 3.17, T and S are inverse isomorphisms of one another. Consequently, q is isomorphic to the Banach envelope of l p (δ) via S • F = J .
We close this section with an easy consequence of Theorem 4.2.
n=1 be a sequence of integers in [2, ∞). Then, for y ∈ F N , That is, B l p (δ) is a norming set for the supremum norm.
Proof Just notice that, by Theorem 4.2, the dual map J * of the inclusion map J of l p (δ) into q is an isomorphism. Question 4.2 Suppose 0 < p < q ≤ 1. Is the q-envelope of a L p -space a L q -space?

Upper bounds for k m : the general case.
This section will be devoted to proving that "near unconditional" bases (including quasi-greedy bases) of p-Banach spaces satisfy the estimate Borrowing the terminology from [1] we say that a basis B of a quasi-Banach space Y is suppression unconditional for constants coefficients (SUCC for short) if there is a constant C such that whenever B ⊆ A and γ is a family of signs.
A crucial ingredient in the study of the vestiges of unconditionality enjoyed by quasi-greedy bases in the setting of nonlocally convex quasi-Banach spaces has been the introduction for m ∈ N of the mth restricted truncation operator U m : Y → Y, defined by where A(m, f ) is the mth greedy set of f (see [1, §3.1]).

Lemma 5.1
If sup n U m < ∞ the basis B is SUCC.
Proof Given A, B subsets of N with B ⊆ A, and γ a family of signs, for 0 < < 1 we have where m is the cardinality of B. This implies that Lemma 5.2 Let B = ( y j ) ∞ j=1 be a basis of a quasi-Banach space Y for which the restricted truncation operators are uniformly bounded. Then there is a constant C such that j∈B b j y j ≤ C f whenever B ⊆ N is finite and |b j | ≤ | y * k ( f )| for every j, k ∈ B.
Proof Assume without loss of generality that y = j∈B b j y j = 0, i.e., λ = max j∈B |b j | > 0. Let Since B is SUCC by Lemma 5.1, and B ⊆ A, [1 Of course, B 0 = N. Let m ∈ N and pick N ∈ N ∪ {0} such that 2 N ≤ m < 2 N +1 . For A ⊆ N with |A| ≤ m we consider the partition (A n ) N n=0 of A given by Note that if j, k ∈ A n for some n ≤ N − 1, then By Lemma 5.2, As for the set A N , we have |A N | ≤ |A| ≤ m < 2 N +1 and Using p-convexity we obtain Hence, Proof It is straightforward from [1, Theorem 3.13] and Theorem 5.1.

An upper bound for k m [X p , l p (ı)], 0 < p < 1.
Now we concentrate on the case when the sequence δ = (d n ) ∞ n=1 in the construction of X p (δ) and X p (δ) * is non-decreasing, so that the function σ defined in (3.1) is convex, i.e., (σ (n + 1) − σ (n)) ∞ n=1 is non-decreasing. Notice that any convex function σ : N → N satisfies the inequality Hence, σ is convex and non-decreasing if and only if Iterating this formula yields that if σ 1 and σ 2 are convex and non-decreasing so is We start by proving some additional properties of the integer intervals J k,n = [σ (n) (k), σ (n) (k + 1)) defined in (3.6).

Lemma 5.3
Let 0 < p ≤ 1 and δ = (d n ) ∞ n=1 be a non-decreasing sequence of integers in [2, ∞). For every k ≥ 1 and every n ∈ N \ {0} we have: Proof (a) follows from Fact 3.19 taking into account that ρ is non-decreasing. In order prove (b) we pick j ∈ I k,n and i ∈ I k+1,n . Since j < i by Fact 3.12, the same argument yields (c) follows from the fact that σ (n) is convex.
Given A ⊆ N finite, let us consider the linear map P A : F N → F N defined by The restriction of P A to l p is the coordinate projection S A [X p , l p ] on the set A with respect to the basis X p . Let us also consider the auxiliary linear operator T A : F N → F N given by Lemma 5.4 Let 0 < p ≤ 1 and δ = (d n ) ∞ n=1 be a sequence of integers in [2, ∞). For any finite set A ⊆ N we have Proof The left hand-side inequality is obvious. As for the inequality on the right, it suffices to note that P A = R • T A where R : p → p is a linear operator such that R(e k ) = x k for every k ∈ N. Fact 3.6 implies that R = 2 1/ p .

be a non-decreasing sequence of integers in
Proof In light of Lemma 5.4 and equation (1.3), it suffices to prove that T A (e k ) ≤ (1 + Γ (m)) 1/ p whenever k ∈ N and |A| ≤ m. Since Lemma 5.5 provides the desired estimate.

Optimality of the upper bound for k m [X p , l p ], 0 < p < 1.
Our next task is to show that the estimate provided by Theorem 5.1 is optimal. Like in [5] we will consider the alternative conditionality constants of a basis B = ( y n ) ∞ n=1 of a quasi-Banach space Y, defined for m ∈ N by These constants depend on the particular ordering we choose for the basis, while Proof We recursively define (u k ) ∞ k=1 and (v k ) ∞ k=1 in l p (δ). We start with u 1 = v 1 = x 1 . Assuming that u k and v k have been constructed for k ∈ N, we define Using Fact 3.5, by induction on k ∈ N we obtain that (i) u k and v k are linear combinations of X (k) p with real scalars, (ii) supp(u k ) ⊆ {1} ∪ [k + 1, σ (k + 1)), and (iii) u k (1) = 1.
Whence, by Proposition 3.1, From Fact 3.11 we get u k = 2 1/ p for every k ∈ N. Now we aim at estimating v k from below. By Fact 3.16 and (ii), (iii) and (iv), for every k ∈ N we have Therefore, by Fact 3.13, u k (k + 1) ≤ 0. Then, applying Fact 3.10, for k ∈ N we get v k+1 Combining this inequality with Lemma 5.5 yields for every m ∈ N. Finally, by (i) and (v), Putting together Propositions 5.1 and 5.2 we can state the following theorem.

Lebesgue constant estimates for Lindenstrauss p-bases
We close this section with an estimate for the performance of the greedy algorithm implemented in the space l p (δ) with respect to the Lindenstrauss p-basis X p (δ). To put this small addition in context, we recall that for m ∈ N, the best m-term approximation error of f ∈ Y with respect to B is given by for all f ∈ Y. This is sometimes referred to as a Lebesgue-type inequality for the greedy algorithm.
The growth of the Lebesgue constants as m increases has been studied in [7,14] in the framework of Banach spaces. As for non-locally convex quasi-Banach spaces, let us point out that, if we put the proof of [1, Theorem 6.2] yields that for a super-democratic basis B of a p-Banach 6 Non-equivalent almost-greedy bases in p and l p (ı), 0 < p ≤ 1.
In this section we give a neat application to the structure of the spaces p , 0 < p ≤ 1, in that they contain an uncountable set of mutually non-equivalent (conditional) almost greedy bases. In fact, these bases are not even permutatively equivalent. We will construct each of these bases in a space isomorphic to p instead of in p itself. Let 0 < p ≤ 1. Given a sequence of integers δ in [2, ∞) and an unbounded sequence of positive integers η = (N k ) ∞ k=1 we consider the following direct sum of finite-dimensional Lindenstrauss p-bases: [2, ∞). Then the sequence Γ is doubling. To be precise, it satisfies

Lemma 6.1 Let δ be a sequence of integers in
Proof It is clear by definition that 2k ≤ σ (k) for every k ∈ N. Then, if n = Γ (m) we have 2m < 2Λ(n + 1) ≤ Λ(n + 2) so that Γ (2m) ≤ n + 1. Theorem 6.1 Let 0 < p ≤ 1, δ be a non-decreasing sequence of integers in [2, ∞) and η an unbounded sequence of positive integers. Then Y p [δ, η] is an almost greedy Schauder basis of a space isomorphic to p . Moreover In the case when k j=1 N k N k+1 for k ∈ N, we also have We emphasize that, as we next show, the sequence η chosen for building the direct sum plays no significant role. We say that a subbasis of a Markushevich basis is complemented if the coordinate projection onto the subspace generated by the subbasis is bounded. Of course, in the lack of unconditionality there are subbases that are not complemented, so it seems hopeless trying to extend to Markushevich bases the Schröder-Bernstein theorem for unconditional bases (see [31,Proposition 2.11]). In this situation the decomposition method comes to our aid. The proof of the decomposition method for Markushevich bases in quasi-Banach spaces stated in Lemma 6.2 is similar to the proof of Pełczyński's decomposition method for Banach spaces (see [26] This way, in light of Theorem 6.1 and Corollary 5.1, we obtain a quasi-greedy basis of p "as conditional as possible."

Example 6.2
Assume there is a > 0 such that that d n ≈ n a for n ∈ N. Then σ (n) ≈ n 1+a for n ∈ N. We infer that there are n 0 ∈ N and 0 < C 1 ≤ C 2 < ∞ such that R = σ (n 0 ) > C 1/a 2 and C 1 n 1+a ≤ σ (n) ≤ C 2 n 1+a for every n ≥ N. Then, for n ≥ n 0 , we infer that log R (Λ(n)) ≈ (1 + a) n , n ≥ 0.
Hence for m large enough, The above example hints at the difficulty of estimating the function Γ associated to a given sequence δ. Our strategy here will be to start from where we want to arrive, i.e., we will see that under a mild condition on Γ , there is a sequence δ whose associated function is (equivalent to) Γ . Iterating the argument we obtain We infer that lim x→∞ H (x + 1) − H (x) = ∞. Consequently, there is a ∈ N such that 2 ≤ M(a + 1) − M(a) and Let b ∈ N be such that 2 b ≤ M(a + 1) − M(a) < 2 b+1 . Define It is routine to check that M 2 ≥ 4 and that the sequence (M k ) ∞ k=1 satisfies (6.1). Then, by Proposition 6.2 there exists a non-decreasing sequence δ of integers in [2, ∞) whose associated function Γ is the left inverse of (M k ) ∞ k=1 . If m ≥ 2 b and Γ (m) = n we have b − a − 1 + φ(log(m + M(a) − 2 b + 1)) < n < b − a + φ(log(m + M(a) − 2 b + 1)).
Since the function φ • log is doubling on the interval [2, ∞), we infer that Γ (m) ≈ φ(log(m)) for m ≥ 2. Proof For l p it follows from Proposition 6.3 and Theorem 5.2, and for p it follows from Proposition 6.3, Theorem 6.1. Proof Just apply Theorem 6.2 with φ(x) = x c , 0 < c ≤ 1.
The following result was proved for the case 1 in [8] using completly different techniques.
Corollary 6.2 Both the spaces l p (δ) and p for 0 < p ≤ 1 contain a continuum of mutually permutatively non-equivalent almost greedy bases.
Proof It is immediate from Corollary 6.1.

Lindenstrauss dual bases
the democracy functions and quasi-greedy constants of X * 1 . The mth Lebesgue (quasigreedy) constant L q m [B, Y] of a basis B of a quasi-Banach space Y will be the smallest constant C such that Lemma 7.1 Let δ be a non-decreasing sequence of integers in [2, ∞). Then X * 1 is not SUCC, therefore it is neither quasi-greedy nor superdemocratic. Quantitatively, for every m ∈ N: Let k ∈ N. By Fact 3.13 the sequence ((−1) j x * j (k)) ∞ j=1 is alternating. Since We infer that X * 1 is not SUCC and that, if C m = L q m [X * 1 , l * 1,0 ], We also provide an estimate for the lower democracy function.