Bulletin of the Australian Mathematical Society

Research Article

On Banach spaces of vector valued continuous functions

Pilar Cembranosa1

a1 Departamento de Teoria de Funciones, Facultad de Ciencias Matematicas, Universidad Complutense de Madrid, Madrid 3, Spain.

Abstract

Let K be 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, 1], 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 Dieudonné 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.

(Received May 23 1983)