A Note on Gowers' Dichotomy Theorem

Summary

We present a direct proof, slightly different from the original, for an important special case of Gowers’ general dichotomy result: If X is an arbitrary infinite dimensional Banach space, either X has a subspace with unconditional basis, or X contains a hereditarily indecomposable subspace.

The first example of dichotomy related to the topic discussed in this note is the classical combinatorial result of Ramsey: for every set A of pairs of integers, there exists an infinite subset M of ℕ such that, either every pair ﹛m₁, m₂﹜ from M is in A, or no pair from M is in the set A. There exist various generalizations to “infinite Ramsey theorems” for sets of finite or infinite sequences of integers, beginning with the result of Nash-Williams [NW]: for any set A of finite increasing sequences of integers, there exists an infinite subset M of N such that either no finite sequence from M is in A, or every infinite increasing sequence from M has some initial segment in A (although it does not look so at the first glance, notice that the result is symmetric in A and A^C, the complementary set of A; for further developments, see also [GPl, [E]). The first naive attempt to generalize this result to a vector space setting would be to ask the following question: given a normed space X with a basis, and a set A of finite sequences of blocks in X (i.e., finite sequences of vectors (x₁, … , x_k) where x1, … , x_k ∈ X are successive linear combinations from the given basis), does there exist a vector subspace Y of X spanned by a block basis, such that either every infinite sequence of blocks from Y has some initial segment in A, or no finite sequence of blocks from Y belongs to A, up to some obviously necessary perturbation involving the norm of X.