On Banach-Spaces Of Vector-Valued Continuous-Functions

Abstract

Let K tie a compact Hausdorff space and let E be a Banach
Space. We denote by C(K, E) the Banach space of all E-valued
Continuous functions defined on K , endowed with the supremum Norm.
Recently, Talagrand [Israel J. Math. 44 (1983), 317-321]
Constructed a Banach space E having the Dunford-Pettis property
Such that C([0, l ] , E) fails to have the Dunford-Pettis property.
So he answered negatively a question which was posed some years ago.
We prove in this paper that for a large class of compacts K
(the scattered compacts), C(K, E) has either the Dunford-Pettis
Property, or the reciprocal Dunford-Pettis property, or the
Dieudonne property, or property V if and only if E has the
Same property.
Also some properties of the operators defined on C(K, E) are
Studied.