Abstract :
We prove that a Banach space X has the metric approximation property if and only if F(Y,X), the space of all finite rank operators, is an ideal in L(Y,X), the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if F(X,Y) is an ideal in L(X,Y) for every Banach space Y.
Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.
