Recent Results in the Theory of Infinite-Dimensional Banach Spaces

Many such questions have been solved in the last three years, and surprising connections between them have been discovered. The purpose of this paper is to explain these developments. Unless otherwise stated, all spaces and subspaces mentioned will be infinite-dimensional separable Banach spaces. We begin by discussing questions of the first kind above. It is clear straight away that not every space has a Hilbert space, the nicest space of all, as a subspace; the space t\ is but one of many obvious counterexamples. However, if one asks whether every space contains Co or £p for some 1 < p < oo, then one already has a very simple question to which the answer is not at all obvious. In fact, this question was not answered until the early 1970s, when Tsirelson [T] used a clever inductive procedure to define a counterexample. The proof that his example does not contain Co or £p is surprisingly short (this is even more true of the dual of his space as presented by Figiel and Johnson [FJ]), but the ideas he introduced have been at the heart of the recent progress. There were two further weakenings of the notion of "nice" that left questions not answered by Tsirelson's example. For the first, recall that a Schauder basis (we will often say simply basis) of a Banach space X is a sequence (xn)^L1 such that every vector in X has a unique expression as a norm-convergent sum of the form Y^=i n%nA simple result proved in the early 1930s by Mazur (see [LT2]) is that every Banach space has a subspace with a basis. Whether every separable Banach space had a basis was a famous open problem, answered negatively by Enfio [En] in 1973. The definition of a Schauder basis is unlike that of an algebraic basis for a vector space in that the order of the xn is important. A basis (xn)TM=l with the property that (^7r(n))^Li is a basis for every permutation 7r of the positive integers is called an unconditional basis. It was shown by Mazur that (xn)^L=1 is