Dual problems for weak and quasi approximation properties

It is shown that for the separable dual X∗ of a Banach space X if X∗ has the weak approximation property, then X has the metric quasi approximation property. Using this it is shown that for the separable dual X∗ of a Banach space X the quasi approximation property and metric quasi approximation property are inherited from X∗ to X and for a separable and reflexive Banach space X, X having the weak approximation property, bounded weak approximation property, quasi approximation property, metric weak approximation property, and metric quasi approximation property are equivalent. Also it is shown that the weak approximation property, bounded weak approximation property, and quasi approximation property are not inherited from a Banach space X to X∗. © 2005 Elsevier Inc. All rights reserved