Abstract
We introduce the properties W ∗ D and BW ∗ D for the dual space of a Banach space. And then solve the dual problem for the compact approximation property (CAP): if X ∗ has the CAP and the W ∗ D , then X has the CAP. Also, we solve the three space problem for the CAP: for example, if M is a closed subspace of a Banach space such that M ⊥ is complemented in X ∗ and X ∗ has the W ∗ D , then X has the CAP whenever X / M has the CAP and M has the bounded CAP. Corresponding problems for the bounded compact approximation property are also addressed.
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