PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 10, October 1996 GOWERS’ DICHOTOMY FOR ASYMPTOTIC STRUCTURE R. WAGNER (Communicated by Dale Alspach) Abstract. In this paper Gowers’ dichotomy is extended to the context of weaker forms of unconditionality, most notably asymptotic unconditionality. A general dichotomic principle is demonstrated; a Banach space has either a subspace with some unconditionality property, or a subspace with a corre- sponding ‘proximity of subspaces’ property. 0. Notation In this paper, unless stated otherwise, all spaces will be infinite dimensional. Once we have a basis {x i } ∞ i=1 in a space X , we define a finite support vector to be a vector of the form ∑ m i=n a i x i . The range of a finite support vector x will be the smallest interval [n, m] such that x may be written as ∑ m i=n a i x i . We write x<y for two finite support vectors x and y if the range of x ends before the range of y begins (i.e. supp(x)=[n 1 ,m 1 ], supp(y)=[n 2 ,m 2 ] and m 1 <n 2 ). We say that the vectors {y i } k i=1 are consecutive if y 1 <y 2 < ··· <y k . A block subspace of X (with respect to a basis) is a subspace generated by a basic sequence of consecutive finite support vectors. Finally, define an H.I. (Hereditarily Indecomposable) space to be a Banach space, in which two infinite dimensional subspaces have zero angle between them (i.e. for every ε> 0 and for all Y,Z ⊂ X , subspaces, there are vectors y ∈ Y , z ∈ Z , ‖y‖ = ‖z ‖ = 1, ‖y − z ‖ <ε). This also means that the span of any two disjoint infinite dimensional closed subspaces is not a closed subspace of X . The existence of such a space was recently proved in [GM]. 1. Introduction This paper is an application of the ideas of [G] to the language of asymptotic structure introduced in [MMiT]. The theorem at hand is the following: Theorem 1.1 (Gowers). Every Banach space X contains a subspace with an un- conditional basis, or an H.I. subspace. This theorem is based on a combinatorial result which concerns the following game. In a space X with a basis define Σ(X )= {{x i } n i=1 ; x i are consecutive Received by the editors March 27, 1995. 1991 Mathematics Subject Classification. Primary 46B20. The author was partially supported by BSF. c 1996 American Mathematical Society 3089