Introduction, definitions and discussion of results.

Although the example given by Enflo in 1973 [5] settled the approximation problem and the basis problem for Banach spaces, a number of closely related problems have continued to arouse interest. If X is a separable Banach space, there are a number of natural properties intermediate between X having the approximation property and having a basis.

Let us first make some definitions. Suppose X is a separable Banach space. Then X has the approximation property (AP) if there is a net of finite-rank operators Tα so that Tαx → x for x ∈ X, uniformly on compact sets. is said to have the bounded approximation property (BAP) if this net can be replaced by a sequence Tn; alternatively X has (BAP) if there is a sequence of finite-rank operators, Tn, such that sup Tn∥ > ∞ and Tnx → x for x ∈ X. A sequence Tn with these properties will be called an approximating sequence. If X has an approximating sequence Tn with limn → ∞ ∥Tn∥ = 1 then X has the metric approximation property (MAP).

An important principle [15] that we will use frequently is that if Tn is any approximating sequence for X then there is an approximating sequence Sn satisfying SmSn = Sn whenever m > n and such that for some subsequence Tkn of Tn then limn → ∞∥Tkn − Sn∥ = 0. (See Lemma 2.4 of [15]).