Introduction

Abstract

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 l_1-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, l_2, is distortable. This is equivalent to the following approximate Ramsey problem. Let S_(l2) = {x ∈ l_2: ∥x∥ = 1} be the unit sphere of l_2. Finitely color the sphere by colors C_1,…, C_k and let ε > 0. Does there exist an i_0 and an infinite dimensional closed linear subspace X of l_2 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 l_2 and confine the search to block subspaces — those spanned by block bases of (e_i) (these terms are defined precisely below).