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On Banach-Spaces Of Vector-Valued Continuous-Functions

Cembranos Diaz, Pilar (1983) On Banach-Spaces Of Vector-Valued Continuous-Functions. Bulletin Of The Australian Mathematical Society, 28 (2). pp. 175-186. ISSN 0004-9727

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Abstract

Let K tie 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, l ] , 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
Dieudonne 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.

Item Type:Article
Uncontrolled Keywords:Mathematics
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:14976
References:

[1] Jurgen Batt and E. Jeffrey Berg, "Linear bounded transformations on the space of continuous functions", J. Funct. Anal. 4 (1969), 215-239.

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

[3] Ivan Dobrakov, "On representation of linear operators on CAT, X) ", Czechoslovak Math. J. 21 (96) (1971), 13-30.

[4] A. Grothendieck, "Sur l e s applications lineaires faiblement compactes d'espaces du type C(K) ", Canad. J. Math. 5 (1953), 129-173.

[5] John Horvath, Topological vector spaces and distributions, Volume I (Addison-Wesley, Reading, Massachusetts; Palo Alto; London; 1966).

[6] Joram Lindenstrauss, Lior Tzafriri, Classical Banach spaces. 1. Sequence spaces (Ergebnisse der Mathematik und ihrer Grenzgebiete, 92. Springer-Verlag, Berlin, Heidelberg, New York, 1977.

[7] A. Pelczynski, "Banach spaces on which every unconditionally converging operator is weakly compact", Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 10 (1962), 61»1-6U8.

[8] Zbigniew Semadeni, Banach spaces of continuous functions (Monografie Matematyczne, 55. PWH - Polish Scientific Publishers, Warszawa, 1971).

[9] M. Talagrand, "La propriete de Dunford-Pettis dans C{K, E) et LX(E) ", Israel J. Math. 44 (1983), 317-321.

Deposited On:24 Apr 2012 11:39
Last Modified:06 Feb 2014 10:13

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