Approximation properties for dual spaces - Lima, Vegard (MATHEMATICA SCANDINAVICA, 2003)
We prove that a Banach space X has the metric approximation property if and only if F(Y,X) is an ideal in L(Y,X^**) for all Banach spaces Y. Furthermore, X^* has the metric approximation property if and only if for all Banach spaces Y and all Hahn-Banach extension operators φ : X^* → X^*** there exists a Hahn-Banach extension operator Φ : F(Y,X)^* → L(Y,X^**)^* such that Φ(x^* y^**) = (φ x^*) y^** for all x^* X^* and all y^** Y^**. We also prove that X^* has the approximation property if and only if for all Banach spaces Y and all Hahn-Banach extension operators φ : X^* → X^*** there exists a Hahn-Banach extension operator Φ : F(Y,X)^* → W(Y,X^**)^* such that Φ(x^* y^**) = (φ x^*) y^** for all x^* X^* and all y^** Y^**, which in turn is equivalent to F(Y,X) being an ideal in W(Y,X^**) for all Banach spaces Y and all equivalent renormings X of X.