1 - Introduction

Summary

The most outstanding problems in the theory of infinite dimensional Banach spaces, those that were central to the study of the general structure of a Banach space, finally yielded their secrets in the 1990's. In this survey we shall discuss these problems and their solutions and more. For many years researchers have been aware of deep connections between both the theorems and ideas of logic and set theory and Banach space theory. We shall try to illuminate these connections as well.

For example the ideas of Ramsey theory played a key role in H. Rosenthal's magnificent l₁-theorem in 1974 [R1]. But there is also a less direct connection with the Banach space question as to whether or not separable infinite dimensional Hilbert space, l₂, is distortable. This is equivalent to the following approximate Ramsey problem. Let S_l2 = {x ∈ l₂: ∥x∥ = 1} be the unit sphere of l₂. Finitely color the sphere by colors C₁,…, C_k and let ε > 0. Does there exist an i₀ and an infinite dimensional closed linear subspace X of l₂ so that the unit sphere of X, S_X, is a subset of (C_i0)ε = {y ∈ S_l2 : ∥y − x∥ < ε for some x ∈ C_i0}? It suffices to let (e_i) be an orthonormal basis for l₂ and confine the search to block subspaces — those spanned by block bases of (e_i) (these terms are defined precisely below).