Ideals of finite rank operators, intersection properties of balls, and theapproximation property

We characterize the approximation property of Banach spaces and their dual spaces by the position of finite rank operators in the space of compact ope rators. In particular, we show that a Banach space E has the approximation property if and only if for all closed subspaces F of c(0), the space F(F, E) of finite rank operators from F to E has the n-intersection property in the corresponding space K(F, E) of compact operators for all n, or equivale ntly, F(F, E) is an ideal in K(F, E).