We cannot end before at least briefly discussing one other spectacular result of the 90's.

We recall that the homogeneous Banach space problem (P5) is: If X is isomorphic to all Y ⊆ X, is X isomorphic to l2? This was solved by combining two beautiful pieces of work, Gowers' dichotomy theorem (Theorem 3.1) and the following theorem of Komorowski and Tomczak-Jaegermann [KT1, KT2]. A nice exposition somewhat simplifying the argument appears in [TJ1].

8.1 Theorem. If X is homogeneous and not isomorphic to l2 then X has a subspace without an unconditional basis.

It then follows that no subspace of X has an unconditional basis and so by the dichotomy theorem plus the fact that X is homogeneous we have that X must be H.I. But in view of the result of [GM1] that an H.I. space is not isomorphic to any proper subspace, this is impossible. Thus the solution of the homogeneous Banach space problem is achieved.

Komorowski and Tomczak-Jaegermann actually prove something stronger. They show

Theorem. Let X be a Banach space not containing a subspace isomorphic to l2. Then X contains a subspace without an unconditional basis.

Even more recently Komorowski and Tomczak-Jaegermann have made substantial progress on (Q4): if all subspaces of X have an unconditional basis is X isomorphic to l2? They proved [KT3].

Theorem. If every subspace of(X ⊕ X ⊕ …)l2 has an unconditional basis then X is isomorphic to l2.

The proof of Theorem 8.2 is well exposed in [TJ1] and we shall not repeat it here. It is interesting to note that spreading models enter at one part of the argument.

In this chapter we introduce the key ingredients of the logic for normed space structures that is described in this paper. These are the positive bounded formulas and the concept of approximate satisfaction of such formulas in normed space structures.

Let L be a signature for a normed space structure ℳ based on (M(s) ∣ s ∈ S). Recall that S has a distinguished element s = Sℝ for which M(s) = ℝ is the sort of real numbers.

We begin considering ℳ from the model theoretic point of view, introducing a formal language based on L and a semantics according to which this language is interpreted in ℳ. In addition to the symbols of the signature L, we also need for each element s of the sort index set S, a countable set of symbols called the variables of sort s.

We begin defining the formal language by introducing the set of terms of L, or L-terms. Each term is a finite string of symbols, each of which may be a variable or a function symbol of L, or one of the symbols (or, which are used for punctuation. In this many-sorted context, each term is associated with a unique sort which indicates its range. The formal definition is recursive.

Definition. An L-term with range of sort s is a string which can be obtained by finitely many applications of the following rules of formation:

• If x is a variable of L of sort s, then x is a term with range of sort s.

• […]

We have not addressed certain important problems that remain unsolved after many years concerning the classical Banach spaces themselves.

(Q13) Let K be a compact metric space. Is every complemented sub-space of C(K) isomorphic to C(L) for some compact metric space L?

It is known that if K is uncountable then C(K) is isomorphic to C[0,1]. If if is countable then C(K) is isomorphic to C(ωωα) for some α < ω1. Every complemented subspace of c0 (isomorphic to C (ω)) is either finite dimensional or isomorphic to c0 ([Pel]). If X is complemented in C[0,1] and X* is nonseparable then X is isomorphic to C[0,1] [R6]. Every quotient of c0 embeds isomorphically into c0 but this does not hold in general for C(ωωα). A discussion of these and related results may be found in [A1, A2, A3, A4], [Gal, Ga2], [Bo2].

The isomorphism types of the complemented subspaces of L1[0,1] remain unclassified.

(Q14) Let X be a complemented (infinite dimensional) subspace of L1[0,1]. Is X isomorphic to L1 or l1?

Every X which is complemented in lp (1 ≤ p < ∞) or c0 is isomorphic to lp or c0. There are known to be uncountably many mutually nonisomorphic complemented subspaces of Lp[0,1] (1 < p < ∞, p ≠ 2) [BRS] and all are known to have a basis [JRZ]. These spaces have been classified as ℒp spaces ([LP], [LR]), provided they are not Hilbert spaces.

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 Sl2 = {x ∈ l2: ∥x∥ = 1} be the unit sphere of l2. Finitely color the sphere by colors C1,…, Ck 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, SX, is a subset of (Ci0)ε = {y ∈ Sl2 : ∥y − x∥ < ε for some x ∈ Ci0}? It suffices to let (ei) be an orthonormal basis for l2 and confine the search to block subspaces — those spanned by block bases of (ei) (these terms are defined precisely below).