Abstract
LetKbe a compact Hausdorff space and letEbe a Banach space. We denote byC(K, E) the Banach space of allE-valued continuous functions defined onK, endowed with the supremum norm.Recently, Talagrand [Israel J. Math.44(1983), 317–321] constructed a Banach spaceEhaving the Dunford-Pettis property such thatC([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 compactsK(the scattered compacts),C(K, E) has either the Dunford-Pettis property, or the reciprocal Dunford-Pettis property, or the Dieudonné property, or propertyVif and only ifEhas the same property.Also some properties of the operators defined onC(K, E) are studied.
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