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On Banach spaces of vector valued continuous functions

  • Pilar Cembranos (a1)

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.

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[1] Jürgen Batt and E. Jeffrey Berg , “Linear bounded transformations on the space of continuous functions”, J. Funct. Anal. 4 (1969), 215239.

[2] J. Diestel and J.J. Uhl Jr, Vector measures (Mathematical Surveys, 15. American Mathematical Society, Providence, Rhode Island, 1977).

[4] A. Grothendieck , “Sur les applications lineaires faiblement compactes d'espaces du type C(K)”, Canad. J. Math. 5 (1953), 129173.

[6] Joram Lindenstrauss , Lior Tzafriri , Classical Banach spaces. I. Sequence spaces (Ergebnisse der Mathematik und ihrer Grenzgebiete, 92. Springer-Verlag, Berlin, Heidelberg, New York, 1977).

[9] M. Talagrand , “La propriété de Dunford-Pettis dans C(K, E) et L1(E)”, Israel J. Math. 44 (1983), 317321.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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