THE DESCRIPTIVE COMPLEXITY OF THE FAMILY OF BANACH SPACES WITH THE BOUNDED APPROXIMATION

We show that the set of all separable Banach spaces that have the bounded approximation property (BAP) is a Borel subset of the set of all closed subspaces of C(Delta), where Delta is the Cantor set, equipped with the standard Effros-Borel structure. Also, we prove that if X is a separable Banach space with a norming M-basis {e(i), e(i)*}(i=1)(infinity) and E-u = (span) over bar {e(i); i is an element of u} for u is an element of Delta, then the set {u is an element of Delta; E-u has a FDD} is comeager in Delta.