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Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Some combinatorial principles for the dyadic tree
  5. 3 Certain definable sets
  6. 4 Tree-families of continuous functions
  7. 5 The case of nearly unconditionality
  8. 6 The case of convex unconditionality
  9. 7 A dichotomy result for more general tree-families
  10. Acknowledgements
  11. References

Suppose that inline image is a normalized family in a Banach space indexed by the dyadic tree S. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree-families. More precisely, assuming that for any infinite chain β of S the sequence inline image is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence inline image is nearly (resp., convexly) unconditional. In the case where inline image is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree T of S such that for any infinite chain β of T, the sequence inline image is unconditional. Finally, in the more general setting where for any chain β, inline image is a Schauder basic sequence, we obtain a dichotomy result concerning the semi-boundedly completeness of the sequences inline image.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Some combinatorial principles for the dyadic tree
  5. 3 Certain definable sets
  6. 4 Tree-families of continuous functions
  7. 5 The case of nearly unconditionality
  8. 6 The case of convex unconditionality
  9. 7 A dichotomy result for more general tree-families
  10. Acknowledgements
  11. References

In a well-known example Maurey and Rosenthal [7] showed that if inline image is a normalized weakly null sequence in a Banach space then we could not expect that inline image admits an unconditional subsequence. Further, Gowers and Maurey [5] exhibited a Banach space not containing any unconditional basic sequence.

Despite the aforementioned constructions there are some positive results where either some special sequences inline image are considered or weaker forms of unconditionality appear. More precisely, Rosenthal proved the following theorem.

Theorem 1.1. Let K be a compact Hausdorff space and let inline image, inline image, be a sequence of non-zero, continuous, characteristic functions. If inline image converges pointwise to zero, then it contains an unconditional basic subsequence.

Although the initial proof uses transfinite induction, the nature of the previous result is purely combinatorial. Indeed, the proof of Theorem 1.1 (cf. [[1], [8]]) can be obtained from the next result which in turn depends on the infinite Ramsey theorem. In the following, if M is an infinite subset of inline image, inline image denotes the set of all infinite subsets of M.

Theorem 1.2. Let inline image be a compact family of finite subsets of inline image. Then for any inline image, there exists inline image such that the family inline image is hereditary (that is, if inline image and inline image, then inline image).

As a matter of fact, it was infinite Ramsey theory which led to a series of positive results. One of them was obtained by Elton [4] (cf. also [8]).

Theorem 1.3. Every normalized weakly null sequence in a Banach space contains a nearly unconditional subsequence.

The notion of nearly unconditionality concerns the unconditional behavior of linear combinations with coefficients bounded away from zero. The precise definition is the following: A normalized sequence inline image in a Banach space is called nearly unconditional if for every inline image there exists inline image such that for any inline image, any scalars inline image and any inline image,

  • display math

Using Ramsey's theory in a very elegant way, Elton proved the following principle from which Theorem 1.3 is obtained.

Theorem 1.4. Let F be a weakly compact subset of the unit ball of c0 and let inline image and inline image be given. Then for every inline image, there exists inline image such that for every inline image, inline image and inline image with inline image there exists inline image such that

  1. inline image [where inline image]
  2. inline image, where inline image.

The subsequences of a weakly null sequence have also been investigated with respect to the property of convex unconditionality. A normalized sequence inline image in a Banach space is called convexly unconditional if for every inline image there exists inline image such that for any absolutely convex combination inline image with inline image and any sequence inline image of signs,

  • display math

The next result, concerning the case of convex unconditionality, has been proved by Argyros, Mercourakis and Tsarpalias [2].

Theorem 1.5. Every normalized weakly null sequence in a Banach space contains a convexly unconditional subsequence.

As in the previous cases, the proof of this theorem is based on the next combinatorial principle.

Theorem 1.6. Let F be a weakly compact subset of c0 and let inline image and inline image be given. Then for every inline image, there exists inline image such that for every inline image, inline image and inline image with inline image there exists inline image satisfying the conditions

  1. inline image
  2. inline image.

Finally, the combinatorial proof of Rosenthal's theorem has been expanded and some stronger results have been obtained. As pointed out in [3], Theorem 1.1 can not be extended in the case where the range of inline image is a finite set of arbitrarily large cardinality. However, Arvanitakis [3] expanded this theorem in the case where the cardinality of the range of inline image is finite and uniformly bounded by some positive integer.

Theorem 1.7. Let K be a Hausdorff compact space, X a Banach space and inline image, inline image, a normalized sequence of continuous functions. We assume that inline image converges pointwise to zero and that the range of inline image's is of finite cardinality uniformly bounded by some positive integer J. Then inline image contains an unconditional subsequence.

The following result, also proved in [3], concerns the case where the space X in the above theorem is finite dimensional.

Theorem 1.8. Let K be a Hausdorff compact space and inline image, inline image, a uniformly bounded sequence of continuous functions which converges pointwise to zero. We also assume that there are a null sequence inline image of positive numbers and a positive real number μ such that for every inline image and any inline image either inline image or inline image. Then inline image contains an unconditional subsequence.

The above theorems are derived by the following combinatorial principle (cf. also [3]) extending Theorem 1.2.

Theorem 1.9. Assume that I is a set, n a positive integer and for any inline image, inline image are finite subsets of inline image such that setting inline image, the closure of the family inline image in the pointwise topology contains only finite sets. Then for any inline image, there exists inline image such that the following holds:

Given inline image, inline image, inline image and inline image then there exists inline image such that inline image and inline image for any inline image.

Throughout this paper S denotes the standard dyadic tree, that is the set inline image of all finite sequences in inline image, including the empty sequence denoted by inline image. The elements inline image are called nodes. If s is a node and inline image, we say that s is on the n-th level of S. We denote the level of a node s by lev(s). The initial segment partial ordering on S is denoted by ⩽ and we write inline image if inline image and inline image. If inline image, we say inline image is a follower of s while if inline image are nodes such that neither inline image nor inline image then s and inline image are called incomparable. We also say that the nodes inline image and inline image are the successors of the node s.

A partially ordered set T is called a dyadic tree if it is order isomorphic to inline image. A subtree of S is any subset inline image of S which has a single minimal element and any element of inline image has exactly two successors. In the sequel we mean by a tree always a dyadic tree. If T is a tree, a chain of T is an infinite linearly ordered subset of T. Throughout this paper, inline image denotes the set of all chains of T. This notation should not be confused with C([0, 1]). The latter stands for the space of continuous functions on [0, 1] and it is used mainly in Section 'Certain definable sets'.

The set inline image is endowed with the relative topology of the product topology of inline image. A sub-basis of this topology consists of the sets inline image and inline image where t varies over the elements of T. Clearly the sets inline image are open and closed. It is also known (cf. [10]) that inline image is a inline image subset of inline image. Therefore (cf. [6]) the topology of inline image is induced by a complete metric.

Considering on inline image the topology described above, Stern [10] proved the following Ramsey-type theorem for the dyadic tree. Furthermore, Stern (cf. also [10]) applied his combinatorial result in the theory of Banach spaces and extended Rosenthal's ℓ1-theorem [9] to the case of tree-families.

Theorem 1.10. Let T be a tree and let inline image be an analytic set of chains. There exists a subtree inline image of T such that either inline image or inline image.

Stern's initial proof uses forcing methods. However, Henson (cf. [8]) observed that the above mentioned theorem follows from some significant results of Ramsey theory.

In this paper, using Stern's theorem we prove in Section 'Some combinatorial principles for the dyadic tree' some combinatorial principles for the dyadic tree. These results combine combinatorial methods with concepts and techniques coming from analysis and extend Theorems 1.2, 1.4, 1.6 and 1.9 mentioned in this introduction.

In Section 'Certain definable sets', given a normalized family inline image of elements of a Banach space indexed by the dyadic tree S, we verify that the set of all chains with a certain property is Borel or at the most analytic or co-analytic. These verifications allows us to apply Stern's theorem and to obtain the results of the next sections. Especially, in Sections 'Tree-families of continuous functions', 'The case of nearly unconditionality' and 'The case of convex unconditionality', we consider normalized tree-families inline image such that for any chain β of S the sequence inline image is weakly null. Our aim is to investigate the unconditional behaviour of the sequences inline image for inline image. More precisely, in Section 'Tree-families of continuous functions', we prove that in some special cases there exists a subtree T of S such that for any chain β of T the sequence inline image is unconditional. These results extend Rosenthal's and Arvanitakis' theorems. In Section 'The case of nearly unconditionality', we show that there always exists a subtree T of S such that all the sequences inline image, inline image, are nearly unconditional. In Section 'The case of convex unconditionality', we also prove the existence of a subtree T of S such that all the sequences inline image, inline image, are convexly unconditional. These results extend Theorems 1.3 and 1.5 respectively.

Finally, in Section 'A dichotomy result for more general tree-families' we consider the more general case where inline image is a normalized tree-family such that for any chain inline image, inline image is a Schauder basic sequence. In this framework we prove the following dichotomy result (cf. Theorem 7.1): there always exists a subtree T of S such that either (a) for any chain inline image, inline image is semi-boundedly complete, or (b) for no chain inline image, inline image is semi-boundedly complete. Recall that a normalized Schauder basic sequence inline image in a Banach space is called semi-boundedly complete if for every sequence inline image, the condition inline image implies that inline image. Furthermore, if we assume that inline image is weakly null for any chain inline image, then we can combine the above dichotomy with the results of Section 'The case of nearly unconditionality' and we obtain the next stronger result (cf. Theorem 7.2): there always exists a subtree T of S such that either (a) for any chain inline image, inline image is semi-boundedly complete, or (b) for any chain inline image, inline image is C-equivalent to the unit vector basis of c0, where inline image is a common constant. It is worth mentioning that the proof of Theorem 7.1 uses analytic sets. Actually, this is the only point where we appeal to the full strength of Stern's theorem. In all the other cases, the sets appearing in the proofs are Borel sets.

In order to prove the main results of Sections 'Tree-families of continuous functions', 'The case of nearly unconditionality' and 'The case of convex unconditionality', we can rely on the combinatorial principles of Section 'Some combinatorial principles for the dyadic tree' and transfer the arguments of [3], [4] and [2] respectively in the more complicated setting of tree-families. However the proofs obtained are quite long and technical and they will not be presented. Instead our approach uses the corresponding results for sequences and Stern's theorem and provides us with short proofs of the theorems contained in this paper. Although we do not use the principles of Section 'Some combinatorial principles for the dyadic tree', we think that they are of independent interest and they point out the underlying combinatorial nature of the main results of this work.

In conclusion, the purpose of the present work is to extend important results from sequences to tree-families. The essential attitude lying in the core of the paper is that in general this passage from sequences to tree-families often has a fundamental effect in analysis, set theory and logic. This passage is not trivial and usually requires new ideas and techniques. We further believe that the ideas contained in our proofs can be applied in more general concepts.

2 Some combinatorial principles for the dyadic tree

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Some combinatorial principles for the dyadic tree
  5. 3 Certain definable sets
  6. 4 Tree-families of continuous functions
  7. 5 The case of nearly unconditionality
  8. 6 The case of convex unconditionality
  9. 7 A dichotomy result for more general tree-families
  10. Acknowledgements
  11. References

In the following bcn(S) denotes the set of all functions inline image such that f is bounded and for any chain β of S the sequence inline image converges to zero. Clearly, bcn(S) is a linear subspace of the space inline image of bounded real functions defined on S. Further, we consider on bcn(S) the topology of pontwise convergence, that is the relative topology of the product topology of inline image. Since S is countable, bcn(S) is metrizable.

In this section, we first prove the following theorem which expands Elton's combinatorial principle.

Theorem 2.1. Suppose that F is a compact subset of bcn(S), with inline image, where inline image and let inline image and inline image be given. Then there exists a subtree T of S which satisfies the following property:

for any inline image, any inline image, any inline image and any inline image with inline image, there exists inline image such that

  1. inline image [where inline image],
  2. inline image, where inline image or inline image.

Roughly speaking, for any chain β of T and any inline image we find a function inline image which preserves the positive ℓ1 mass of f on the finite set I and is very close to zero on the other inline image's. If we could find a function inline image such that inline image for inline image and inline image for inline image, inline image, then it would follow (cf. [8]) that for any normalized family inline image so that inline image is weakly null for any inline image there is a subtree T of S such that inline image is unconditional for any inline image, which of course is not true.

Proof of Theorem 2.1. Let F0 be a countable dense subset of F. Consider the set inline image of all chains inline image which satisfy the following property: for any inline image, any inline image and any inline image with inline image, inline image, and inline image, there is inline image such that

  1. inline image,
  2. inline image, where inline image or inline image.

Claim 2.2. The set inline image is a Borel subset of inline image.

Indeed, we have

  • display math

where

  • display math
  • display math

Clearly, inline image and inline image are open subsets of inline image; therefore inline image is a Borel set.

Now Stern's theorem implies that there is a subtree T of S such that either (a) inline image or (b) inline image. However, the case (b) can be excluded. Indeed, let us assume that inline image and let β be any chain of T. Applying Theorem 1.4, we find a subchain inline image of β such that for any inline image, any inline image and any inline image with inline image, there is inline image such that inline image and inline image, where inline image or inline image. Since F0 is dense in F, it follows that we can find inline image satisfying the above properties. Hence, inline image and we have reached a contradiction. Therefore, inline image.

Since F0 is dense in F, we can easily verify that the subtree T satisfies the conclusion of the theorem. inline image

In a similar method we also prove the next combinatorial theorem for the dyadic tree.

Theorem 2.3. Suppose that F is a compact subset of bcn(S) which is bounded with respect to the supremum norm and let inline image and inline image be given. Then there exists a subtree T of S satisfying the following property:

for any chain inline image of T, any inline image, any inline image and any inline image with inline image, there exists inline image such that

  1. inline image
  2. inline image.

Proof. Let F0 be a countable dense subset of F and let inline image be the set of all chains inline image which satisfy the following: for any inline image, any inline image and any inline image with inline image there is inline image such that inline image and inline image. Then,

  • display math

where

  • display math
  • display math

It follows that inline image is a Borel subset of inline image. Stern's theorem implies that there is a subtree T of S such that either (a) inline image or (b) inline image. By Theorem 1.6, the case (b) is excluded. Therefore, inline image and the result follows. inline image

Finally, we expand Theorem 1.9 to obtain a combinatorial theorem for trees.

Theorem 2.4. Assume that I is a set, n a positive integer and for every inline image, inline image are finite subsets of S. For any inline image we set inline image and let inline image. We also assume that for any F in the closure of inline image and for any chain inline image, the set inline image is finite. Then there exists a subtree T of S satisfying the following property:

for any chain inline image, any inline image, any inline image, any inline image and any inline image there exists inline image such that

  1. inline image
  2. inline image for any inline image.

Proof. The powerset inline image endowed with the product topology is a compact metric space. Therefore, inline image is separable. It follows that for any inline image there is a countable subset inline image such that inline image is a dense subset of inline image. We set inline image. Clearly, J is countable and for every inline image, inline image is dense in inline image.

We consider the set inline image of all chains inline image which satisfy the following property: for any inline image, any inline image, any inline image and any inline image there is inline image such that inline image and inline image for any inline image. Then we have

  • display math

where

  • display math

The sets inline image and inline image are open; therefore inline image is a Borel subset of inline image. By Stern's theorem, there is a subtree T of S such that either (a) inline image or (b) inline image. By Theorem 1.9 we must have (a). Since inline image is dense in inline image for any inline image, the tree T satisfies the desired property. inline image

3 Certain definable sets

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Some combinatorial principles for the dyadic tree
  5. 3 Certain definable sets
  6. 4 Tree-families of continuous functions
  7. 5 The case of nearly unconditionality
  8. 6 The case of convex unconditionality
  9. 7 A dichotomy result for more general tree-families
  10. Acknowledgements
  11. References

The results of this paper are based on applications of the Ramsey-type theorem of Stern for trees. However, to apply Stern's theorem (cf. Theorem 1.10) it is necessary first to verify that given a tree-family inline image in a Banach space, the set of all chains inline image that satisfy a certain property is a definable set, i.e., it is Borel or at the most analytic or co-analytic. We explain here the idea on which the verifications of that sort are based. Each of the properties of interest is described exclusively by conditions with finitely many parameters that are running in countable sets and the quantifiers “for all” and “exists”. If one considers the relevant concept in this way, it is easily understood that the sets which appear in the corresponding theorems are Borel and in only one case we take after all the quantifiers a continuous projection and finally the complement of its image, hence the resulting set is co-analytic. Nevertheless, for the proofs to be completely exact the technical details are rather complicated mainly because of the inevitable long symbolism. However, having in mind the idea we previously described, one can easily read the exact proofs.

Initially, we limit ourselves in subsets of the Polish space inline image that are described as above, and then using a suitable mapping we transfer to the space of chains. Characteristically we give a detailed proof in a certain case that the corresponding set is Borel, while in the other cases we quote only the description of the corresponding sets. More precisely, we define the following subsets of inline image:

  • display math

Then, we have the next proposition.

Proposition 3.1. The sets inline image, inline image, inline image, inline image are Borel subsets of inline image, while inline image is a co-analytic subset of inline image.

Proof. We prove in detail that inline image is a Borel set. Firstly, the property “inline image is a normalized sequence” is obviously closed and we shall not refer to it in what follows. We observe now that for any constant inline image, inline image, inline image and inline image the set

  • display math

is a closed subset of inline image. We also have that:

  • display math

Therefore

  • display math

and obviously inline image is a Borel set.

Similarly, we can see that

  • display math

Hence, inline image is a Borel set.

In order to prove that inline image is Borel, we set

  • display math

where inline image, inline image, inline image, inline image and inline image. The sets of the form inline image are closed, while the sets of the form inline image are open. Moreover, we have

  • display math

and this implies that inline image is a Borel set.

Similarly, for the set inline image, we consider the following

  • display math

where inline image, inline image, inline image, inline image. For constant values of the parameters the sets inline image and inline image are closed subsets of inline image. Hence, inline image is a Borel set, as it is written as follows

  • display math

Finally, as far as the set inline image, we consider the following set

  • display math

It is easily verified that inline image is a Borel subset of the product inline image. Hence, inline image is a co-analytic set being the complement of the first projection of inline image. inline image

We assume now that inline image is a normalized tree-family in a Banach space X. We consider the subspace Y generated from this family, i.e., inline image. The space Y is a separable Banach space. It is well-known that any separable Banach space embeds isometrically in C([0, 1]). Therefore, Y can be viewed as a subspace of C([0, 1]). We now define the mapping inline image which in every chain inline image associates the sequence inline image as an element of inline image. Obviously, φ is an embedding. Further, we set

  • display math

Clearly, inline image, inline image, inline image, inline image, inline image are the inverse images via the map φ of the sets inline image, inline image, inline image, inline image, inline image respectively. Therefore, by Proposition 3.1, we immediately have the following corollary.

Corollary 3.2. The sets inline image, inline image, inline image, inline image are Borel subsets of inline image and inline image is a co-analytic subset of inline image.

4 Tree-families of continuous functions

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Some combinatorial principles for the dyadic tree
  5. 3 Certain definable sets
  6. 4 Tree-families of continuous functions
  7. 5 The case of nearly unconditionality
  8. 6 The case of convex unconditionality
  9. 7 A dichotomy result for more general tree-families
  10. Acknowledgements
  11. References

In this section we consider tree-families inline image of continuous functions. Then, under some conditions, we show that there exists a subtree T of S such that for any chain inline image, the sequence inline image is unconditional. First, we prove the following general result.

Theorem 4.1. Let inline image be a family in a Banach space X such that inline image, inline image. Suppose that for any chain β of S the sequence inline image contains an unconditional subsequence. Then there exist a subtree T of S and a constant inline image such that for any chain β of T, the sequence inline image is C-unconditional.

Proof. We consider the set inline image. By Corollary 3.2, we have that inline image is a Borel subset of inline image. Stern's theorem implies now that there exists a subtree inline image of S such that either (a) inline image or (b) inline image. By our hypotheses, if β is any chain of inline image, then there is a subchain inline image such that inline image is an unconditional sequence. Therefore, inline image and the case (b) is impossible. Hence, we have that inline image, that is for any chain β of inline image the sequence inline image is unconditional.

It remains to prove that we can find a subtree T of inline image such that the sequences inline image, inline image, share the same unconditional constant C. To avoid introducing additional notation, we assume that for any chain β of the original tree S, inline image is unconditional, that is inline image. Then, we have:

  • display math

where inline image is the set of all chains β such that inline image is C-unconditional. Following the ideas of Section 'Certain definable sets' it is easy to verify that inline image is a closed set and, hence, inline image is an inline image set. Therefore, the Baire category theorem implies that there exists a constant C such that the set inline image has non-empty interior. This means that there are finitely many nodes inline image such that for any chain β beginning with inline image, the sequence inline image is C-unconditional. Let T be the subtree consisting of the node inline image and all its followers. Clearly, inline image.

inline image

Combining Theorem 4.1 with Theorems 1.7 and 1.8, we obtain the following results.

Theorem 4.2. Let K be a Hausdorff compact space, X a Banach space and inline image, inline image, a normalized family of continuous functions. We assume that for any maximal chain inline image, the sequence inline image converges pointwise to zero and that there exists a positive integer inline image such that inline image for any inline image. Then there exist a subtree T of S and a constant inline image such that for any chain β of T, the sequence inline image is C-unconditional.

Theorem 4.3. Let K be a Hausdorff compact space and let inline image, inline image be a family of continuous functions. We assume that for any maximal chain β of S, the sequence inline image is uniformly bounded and converges pointwise to zero. Furthermore, we assume that there are a null sequence inline image of positive real numbers and a constant inline image such that for any inline image and any inline image either inline image or inline image. Then there exist a subtree T of S and a constant inline image such that for any chain β of T, inline image is a C-unconditional sequence.

5 The case of nearly unconditionality

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Some combinatorial principles for the dyadic tree
  5. 3 Certain definable sets
  6. 4 Tree-families of continuous functions
  7. 5 The case of nearly unconditionality
  8. 6 The case of convex unconditionality
  9. 7 A dichotomy result for more general tree-families
  10. Acknowledgements
  11. References

In this section we prove the analogous to Elton's theorem for the case of tree-families. Further, as in Theorem 4.1, we obtain a uniformity of the constants on the chains. More precisely, we have the following.

Theorem 5.1. Let inline image be a normalized family in a Banach space X. Assume that for every chain inline image, the sequence inline image is weakly null. Then there exists a subtree T of S with the following property: for every inline image there exists inline image such that for any chain inline image of T, any inline image, any inline image and any inline image,

  • display math

That is, for any chain inline image, the sequence inline image is nearly unconditional and the constant inline image is independent of the chain β.

It is well-known that any normalized weakly null sequence in a Banach space contains a Schauder basic subsequence. The proof of this result can be easily transferred to tree-families. Thus we obtain the next lemma whose proof is omitted.

Lemma 5.2. Suppose that inline image is a normalized family in a Banach space, such that for any chain β of S the sequence inline image is weakly null. Then, for every inline image there exists a subtree T of S such that for any chain β of T, inline image is inline image-basic.

Proof of Theorem 5.1. We may assume, by passing to a subtree if necessary, that for any chain β of S, inline image is a basic sequence with basis constant inline image, where D is an absolute constant. We next consider the set inline image. By Corollary 3.2, we know that inline image is a Borel subset of inline image. Therefore, Stern's theorem implies that there is a subtree inline image of S such that either (a) inline image or (b) inline image. By Theorem 1.3 every normalized, weakly null sequence contains a nearly unconditional subsequence; therefore the case (b) is impossible. Thus inline image, that is for any chain β of inline image the sequence inline image is nearly unconditional.

Now assume that for the original tree S we have inline image. It remains to show that there is a subtree T of S such that for every inline image there is inline image such that for any chain inline image of T, any inline image, any inline image and any inline image, inline image. That is, the constant C is independent of β.

We start with the following observation. We fix some positive number δ. Since for any inline image, inline image is nearly unconditional, it follows that

  • display math

where, for fixed inline image and inline image, inline image denotes the set of all chains inline image such that for any inline image, any inline image and any inline image, inline image. It is easy to verify that inline image is a closed subset of inline image. Therefore, as in the proof of Theorem 4.1, the Baire category theorem implies that there exist a positive constant inline image and a subtree T of S such that inline image. Therefore the subtree T satisfies the desired property, however for the specific number δ.

In order to obtain the general result for arbitrary inline image, we consider a null sequence inline image and we apply a diagonal-type argument for the dyadic tree. The desired subtree T is constructed inductively. We quote the first steps.

Let δ be equal to 1. By the previous observation there are a subtree inline image of S and a positive constant R(1) such that for any chain inline image, any inline image, any inline image and any inline image, inline image. Let inline image be the minimum element of inline image and inline image the nodes placed on the first level of inline image. Then inline image is the minimum node of T and inline image complete the first level of T.

Let δ be equal to 1/2 and let inline image be the subtree of inline image which contains the node t0 and all its followers in inline image. By the previous observation, we find a subtree inline image and a constant inline image such that for any chain inline image of T0, any inline image, any inline image and any inline image, inline image. Let inline image be the nodes placed on the first level of T0. Then inline image are the successors of t0 in T.

The subtree inline image, the constant inline image and the nodes inline image are defined in a similar way. We also set inline image and the second level of T has been completed.

We inductively construct a subtree T of S and positive constants inline image, inline image, satisfying the following property: for any chain inline image of T with inline image, any inline image, any inline image and any inline image

  • display math

Claim 5.3. The subtree T satisfies the conclusion of the theorem.

For any level inline image, let inline image be an enumeration of the maximal linearly ordered subsets of T which contain nodes of level less or equal to r. We set inline image, where inline image is the unconditional constant of the finite sequence inline image. Therefore, for any maximal chain inline image of T, any inline image and any scalars inline image we have

  • display math

Suppose now that k is a positive integer. We show that there is a constant inline image depending only on k such that for any chain inline image, any inline image, any inline image and any inline image, inline image. It suffices to consider only the maximal chains of T. So, if inline image is maximal then

  • display math

The choice inline image completes the proof. inline image

6 The case of convex unconditionality

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Some combinatorial principles for the dyadic tree
  5. 3 Certain definable sets
  6. 4 Tree-families of continuous functions
  7. 5 The case of nearly unconditionality
  8. 6 The case of convex unconditionality
  9. 7 A dichotomy result for more general tree-families
  10. Acknowledgements
  11. References

We now use the techniques of the previous sections in the case of convex unconditionality. As a result, we prove the analogous to Theorem 1.5 for tree-families.

Theorem 6.1. Let inline image be a normalized tree-family in a Banach space X. Assume that for each chain inline image, the sequence inline image is weakly null. Then, there exists a subtree T of S with the following property: for every inline image there exists a constant inline image such that for any chain inline image of T, any absolutely convex combination inline image with inline image and any sequence inline image of signs,

  • display math

That is, for any chain inline image, inline image is a convexly unconditional sequence and the constant inline image depends only on δ.

Proof. The proof follows the lines of the proof of Theorem 5.1. We assume that for any chain β of S the sequence inline image is D-basic. We consider the set inline image. By Corollary 3.2, inline image is a Borel subset of inline image. Stern's theorem implies that there is a subtree inline image of S such that either (a) inline image or (b) inline image. By Theorem 1.5 we must have (a), that is for any chain β of inline image the sequence inline image is convexly unconditional.

We next assume that for the original tree S we have inline image and we show that there is a subtree T of S such that for any inline image there exists inline image depending only on δ such that for any chain inline image of T, any absolutely convex combination inline image with inline image and any sequence inline image of signs, inline image.

For a fixed inline image, as in the proof of Theorem 4.1, we find a subtree which satisfies the desired property for the specific δ. We next consider the sequence inline image and using repeatedly the previous observation we inductively construct a subtree T of S and positive constants inline image, inline image, such that for any inline image, any chain inline image of T with inline image, any absolutely convex combination inline image with inline image and any inline image, inline image.

Claim 6.2. The subtree T satisfies the conclusion of the theorem.

Let inline image. We show that there is a constant inline image depending on k such that for any maximal chain inline image, any absolutely convex combination inline image with inline image and any sequence inline image of signs, inline image. We distinguish the following two cases.

Case 1. Suppose that inline image. Then we have

  • display math

where the constant inline image has been defined in the proof of Theorem 5.1.

Case 2. Suppose that inline image. We set inline image and inline image. Then inline image and x is an absolutely convex combination of inline image such that inline image. By the construction of T, it follows that inline image. Therefore

  • display math

The choice inline image completes the proof. inline image

7 A dichotomy result for more general tree-families

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Some combinatorial principles for the dyadic tree
  5. 3 Certain definable sets
  6. 4 Tree-families of continuous functions
  7. 5 The case of nearly unconditionality
  8. 6 The case of convex unconditionality
  9. 7 A dichotomy result for more general tree-families
  10. Acknowledgements
  11. References

In this section, we consider the more general setting, where inline image is a normalized tree-family such that for any chain β of S, inline image is a Schauder basic sequence. For such families we prove the following dichotomy theorem.

Theorem 7.1. Let inline image be a normalized tree-family in a Banach space X such that for any chain inline image, the sequence inline image is Schauder basic. Then there exists a subtree T of S such that either

  1. for any chain inline image, the sequence inline image is semi-boundedly complete; or
  2. for no chain inline image, the sequence inline image is semi-boundedly complete.

Proof. We consider the set inline image is a semi-boundedly complete basic sequence }. By Corollary 3.2 we know that inline image is a co-analytic subset of inline image and, hence, we can apply Stern's theorem. It follows that there exists a subtree T of S such that either inline image or inline image, that is either

  1. for any chain inline image, the sequence inline image is semi-boundedly complete; or
  2. for no chain inline image, the sequence inline image is semi-boundedly complete.

inline image

Suppose now that inline image is a normalized tree-family such that for any chain inline image, inline image is weakly null. In this case, using Theorem 5.1, we can improve the result of Theorem 7.1 and we obtain the following dichotomy.

Theorem 7.2. Let inline image be a normalized tree-family in a Banach space X. We assume that for any chain inline image, the sequence inline image is weakly null. Then there exists a subtree T of S such that either

  1. for any chain inline image, the sequence inline image is semi-boundedly complete; or
  2. for any chain inline image, the sequence inline image is C-equivalent to the unit vector basis of c0, where inline image is a common constant.

Proof. We may assume that for any chain inline image, the sequence inline image is D-basic. In view of Theorem 7.1, it suffices to consider only the case where no sequence inline image, inline image, is semi-boundedly complete and then we have to prove that there exists a subtree T of S such that for any chain inline image, inline image is equivalent to the unit vector basis of c0. Furthermore, by Theorem 5.1 we may assume that for any chain inline image, the sequence inline image is nearly unconditional.

Our first step is to show that any chain β of S contains a subchain inline image, such that inline image is equivalent to the unit vector basis of c0. The proof of this fact is essentially contained in [8]. However we shall give a brief description.

Since inline image is not semi-boundedly complete, it follows that there exists a bounded sequence inline image not converging to 0 so that inline image. Clearly, we may assume that inline image for every inline image. Let inline image and inline image be such that inline image for all inline image. Then we set inline image and we claim that inline image is equivalent to the unit vector basis of c0.

Firstly, by Theorem 5.1, we have

  • display math

and further, for any signs inline image,

  • display math

(where the constant inline image is given by Theorem 5.1). Therefore, for any inline image, any inline image and any inline image, inline image, we obtain

  • display math

It follows that

  • display math

So far we have shown that any chain inline image contains a subchain inline image such that inline image is equivalent to the unit vector basis of c0. Now, we proceed as follows. We consider the set inline image is equivalent to the unit vector basis of inline image. By Corollary 3.2 the set inline image is a Borel subset of inline image. Therefore, by Theorem 1.10 there exists a subtree inline image of S such that either (a) inline image or (b) inline image. However, the case (b) must be excluded, since for any chain inline image the sequence inline image contains a subsequence equivalent to the basis of c0. Finally, as in the proof of Theorem 4.1, an application of the Baire category theorem shows that we can find a further subtree T of inline image and a constant inline image such that for any chain inline image, inline image is C-equivalent to the unit vector basis of c0. inline image

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Some combinatorial principles for the dyadic tree
  5. 3 Certain definable sets
  6. 4 Tree-families of continuous functions
  7. 5 The case of nearly unconditionality
  8. 6 The case of convex unconditionality
  9. 7 A dichotomy result for more general tree-families
  10. Acknowledgements
  11. References
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    E. Odell, Applications of Ramsey theorems to Banach space theory, in: Notes in Banach spaces, edited by H. E. Lacey (University of Texas Press, 1980), pp. 379404.
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    H. P. Rosenthal, A characterization of Banach spaces containing ℓ1, Proc. Nat. Acad. Sci. USA 71, 24112413 (1974).
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