The hereditary Dunford-Pettis property on C(K,E) (Q1822310)

A Banach space E is said to have the hereditary Dunford-Pettis property if all of its closed subspaces have the Dunford-Pettis property. In this paper we study this property on C(K,E), the Banach space of all continuous functions defined on a compact Hausdorff space K with values in a Banach space E, endowed with the supremum norm. The main result assures that if K is infinite then C(K,E) has the hereditary Dunford-Pettis property if and only if C(K) has the same property and E verifies a certain conditions. This condition means that E has the hereditary Dunford-Pettis property in an uniform sense and it is related to the fact of containing \(c_ 0\) ''uniformly''. We do not know if all Banach spaces with the hereditary Dunford-Pettis property must satisfy such condition, but we note that this is true for most of the known hereditarily Dunford-Pettis spaces.