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On Banach spaces of vector valued continuous functions

Published online by Cambridge University Press:  17 April 2009

Pilar Cembranos
Affiliation:
Departamento de Teoria de Funciones, Facultad de Ciencias Matematicas, Universidad Complutense de Madrid, Madrid 3, Spain.
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1983

References

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