THE Kup-APPROXIMATION PROPERTY AND ITS DUALITY

We introduce an approximation property (K-up-AP, 1 <= p < infinity), which is weaker than the classical approximation property, and discover the duality relationship between the K-up-AP and the K-p-AP. More precisely, we prove that for every 1 < p < infinity, if the dual space X* of a Banach space X has the K-up-AP, then X has the K-p-AP, and if X* has the K-p-AP, then X has the K-up-AP. As a consequence, it follows that every Banach space has the K-u2-AP and that for every 1 < p < infinity, p not equal 2, there exists a separable reflexive Banach space failing to have the K-up-AP.