Notes on approximation properties in separable Banach spaces

Summary

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 T_α so that T_αx → 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 T_n; alternatively X has (BAP) if there is a sequence of finite-rank operators, T_n, such that sup T_n∥ > ∞ and T_nx → x for x ∈ X. A sequence T_n with these properties will be called an approximating sequence. If X has an approximating sequence T_n with lim_{n → ∞} ∥T_n∥ = 1 then X has the metric approximation property (MAP).

An important principle [15] that we will use frequently is that if T_n is any approximating sequence for X then there is an approximating sequence S_n satisfying S_mS_n = S_n whenever m > n and such that for some subsequence T_kn of T_n then lim_{n → ∞}∥T_kn − S_n∥ = 0. (See Lemma 2.4 of [15]).