Schauder bases and the bounded approximation property in separable Banach spaces

Let E be a separable Banach space with the A-bounded approximation property. We show that for each epsilon > 0 there is a Banach space F with a Schauder basis such that E is isometrically isomorphic to a 1-complemented subspace of F and, moreover, the sequence (T(n)) of canonical projections in F has the properties sup(n is an element of N) parallel to T(n)parallel to <= lambda +is an element of and lim sup(n ->infinity) parallel to T(n)parallel to <= lambda This is a sharp quantitative version of a classical result obtained independently by Pelczynski and by Johnson, Rosenthal and Zippin.