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On Banach spaces of vector valued continuous functions

  • Pilar Cembranos (a1)
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.

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References
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[1] Batt Jürgen and Berg E. Jeffrey, “Linear bounded transformations on the space of continuous functions”, J. Funct. Anal. 4 (1969), 215239.
[2] Diestel J. and Uhl J.J. Jr, Vector measures (Mathematical Surveys, 15. American Mathematical Society, Providence, Rhode Island, 1977).
[3] Dobrakov Ivan, “On representation of linear operators on C 0(T, X)Czechoslovak Math. J. 21 (96) (1971), 1330.
[4] Grothendieck A., “Sur les applications lineaires faiblement compactes d'espaces du type C(K)”, Canad. J. Math. 5 (1953), 129173.
[5] Horváth John, Topological vector spaces and distributions, Volume I (Addison-Wesley, Reading, Massachusetts; Palo Alto; London; 1966).
[6] Lindenstrauss Joram, Tzafriri Lior, Classical Banach spaces. I. Sequence spaces (Ergebnisse der Mathematik und ihrer Grenzgebiete, 92. Springer-Verlag, Berlin, Heidelberg, New York, 1977).
[7] Pelczyński A., “Banach spaces on which every unconditionally converging operator is weakly compact”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 10 (1962), 6416148.
[8] Semadeni Zbigniew, Banach spaces of continuous functions (Monografie Matematyczne, 55. PWN – Polish Scientific Publishers, Warszawa, 1971).
[9] Talagrand M., “La propriété de Dunford-Pettis dans C(K, E) et L 1(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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