An Analysis upon the Asymptotic Structure of Banach Spaces: A Case Study of Envelope Functions

INTRODUCTION

Asymptotic structures of infinite-dimensional Banach spaces, introduced in the study of B. Maurey et al. (1994), reflect the behavior at infinity of finite-dimen-sional subspaces which repeatedly appear everywhere and far away in the space and are arbitrarily spread out along, for instance, a basis. This approach to infinite-dimensional spaces serves as a bridge between finite-dimensional and infinite-dimensional theories, in view of the outstanding developments in the Banach space theory in the 1990?s. For example, asymptotic- spaces were discovered in V.D. Milman and N. Tomczak-Jaegermann (1993) in connection with the distortion problem; and the game approach used in the study of B. Maurey et al. (1994) to define asymptotic structures originated by W.T. Gowers(2002). For these and many other aspects of asymptotic approaches to infinite-dimensional Banach spaces theory we refer the reader to the exhaustive survey by E. Odell (2002). In its simplest form, the asymptotic structure of a Banach space is defined as follows. Given a Banach space X with a monotone basis, an n- dimensional space E with a monotone basis is an asymptotic space for X if there exists a finitely supported normalized vector y1 (block) with support arbitrarily far along the basis , then a normalized block y2with support arbitrarily far after the support, of y1, then a normalized block y3 with support arbitrarily far after the support of y2 and so on, such that the blocks obtained after n steps have behavior as close to the behavior of as we wish. (This means that any linear combination of has norm in X arbitrarily close to the norm in E of the corresponding linear combination of The normalized blocks are called permissible vectors. The set of all n-dimensional asymptotic spaces of X will be denoted by . The asymptotic structure of X consists of all for all A Banach space X is called an asymptotic- space if there exists a constant such that for all n and the basis in E is C-equivalent to the unit vector basis of (for the precise definition see below). That is, asymptotic- spaces have only one type of asymptotic spaces. In [MMT], it is

is C-isomorphic to then X is asymptotic- This means that in such situations, there is a natural isomorphism between E and which is the equivalence between respective bases. Up to a constant, for asymptotic- spaces have a unique asymptotic structure, and this in fact characterizes asymptotic- spaces. The notion of disjoint-envelope functions is a convenient tool for studying asymptotic structures of spaces with an unconditional basis (or more generally, with asymptotic unconditional structure). In general, asymptotic methods in the theory of infinite-dimensional Banach spaces rely on stabilizing information of finite nature \at infinity". This way we discard properties which may appear sporadically in the space and could be removed by passing to appropriately chosen subspaces or some other substructures. First methods of this kind began to develop in the 1970's, in connection with Ramsey theorems and the notion of a spreading model (to be described later in this introduction). The ideas behind what is now called infinite-dimensional Banach spaces. From the very beginning of Functional Analysis initiated by the work of Banach in the 1920's the objective of the classical theory of infinite-dimensional spaces have been mainly to establish a structure theory for Banach spaces. Besides isomorphism type questions, primarily, the problems were centered around seeking subspaces with `nice' structural properties in all Banach spaces.

In this study, all spaces are real separable Banach spaces and all subspaces are closed subspaces. By X, Y,... we usually denote infinite-dimensional Banach spaces; we reserve E. F,... to denote finite-dimensional Banach spaces. The norm in X is denoted by ||.||x, or simply by ||.|| if there is no ambiguity. By Bx we denote the closed unit ball , and by the unit sphere of X. Linear continuous maps between two Banach spaces X and Y are called operators and denoted by . If T is an isomorphism between X and Y, the isomorphism constant C is defined by and in this case we write or simply if we do not want to specify the isomorphism constant. We will say that X and Y are C-isomorpliic or simply isomorphic. For a set E in A, denotes the closed linear span of E in X and conv[£r the closed convex hull of E. As examples of Banach spaces we shall often use the classical sequence spaces and is the space of all real sequences with with the norm For any is the space of real sequences with and the norm is the space of all bounded real sequences with the norm For each denotes the n-

Let X be a Banach space with an asymptotic unconditional structure (with a constant ). We define the set of all normalized disjoint-permissible vectors in X as follows. For if there exist for some and a disjoint partition of First we make a few remarks about the set (where superscript d stands for ‗disjoint‘). Clearly, for all n e N and we have that i.e., If then is an unconditional basic sequence (with constant C). It is also clear that if is a block (successive or just disjoint) basis of some then as well. Finally, if then where is a permutation of This property, obviously, is not shared, in general, by the bases of asymptotic spaces. We also have the following property of which is inherited from If and are in then there exists such that and Indeed, if and are disjoint blocks of the bases and of some asymptotic spaces respectively, then we can find an asymptotic space such that and. Hence the corresponding disjoint blocks of have the desired property. When and are in to avoid repetitions, we will simply say that without referring to We define now the natural analogs of envelope functions on Definition 1 Let X be a Banach space with, an asymptotic unconditional structure. For let and where the inf and the sup is taken over all We call and the disjoint-lower and disjoint-upper-envelope functions respectively. It is easy to see that both functions and are 1-symmetric and 1- unconditional. Moreover, while defines a norm on e0o, satisfies triangle inequality on disjointly supported vectors (of ). Indeed, let and be two disjoint vectors in coo and let be arbitrary. Pick and in such that and Then, by the above remark, and hence Since was arbitrary, it follows that whenever are disjointly supported. To compare the disjoint-envelope functions with the original envelopes, note that for all by the definition of these functions we have, We will use the following convenient notation. Let be the unit vector basis of For occasionally we will write instead of Moreover, for any finite number of successive vectors such that for and for any vector we write instead of where are blocks of the basis of normalized with respect to We‘ll use similar notation for as well.

Let A be an infinite dimensional Banach space. We want to look at finite dimensional subspaces which occur ;at infinity‘. The picture that the reader should have in mind is that of a Banach space with a basis, and the asymptotic spaces as being those finite block subspaces which occur arbitrarily far down the basis. We will define asymptotic spaces using a vector game between two players V and S. Let be a family of infinite dimensional subspaces of X satisfying the filtration condition For example, we could take to be the finite co-dimensional subspaces of A, or if A has a basis we could take the family of tail subspaces. Throughout this study we will be looking at Banach spaces with a basis, so we will assume that the family is the collection of tail subspaces from now onwards. The game is played as follows. S chooses a subspace V responds by choosing a normalized vector S then chooses a subspace with V then responds by choosing a normalized vector such that is basic with basis constant The game continues in this vein until the nth turn, where S selects a subspace with and V chooses a normalized vector such that is basic with basic constant are equivalent to We will call E an asymptotic space of A, if V has a winning strategy for the game for E and every The collection of all 71-dimensional asymptotic spaces for A will be denoted by It is a consequence of Krivine‘s theorem that there exists such that for every n, (with its standard basis) is in Hence, the simplest possible asymptotic structure that can occur is where the basis of every is C-equivalent to the standard basis of for some C independent of n. We will call such spaces asymptotic spaces.

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