Publication: Spaces of vector-valued continuous functions with the Dunford-Pettis property. (Spanish: Espacios de funciones continuas vectoriales con la propiedad de Dunford-Pettis).
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1986
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Real Academia de Ciencias Exactas, Físicas y Naturales
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Abstract
A Banach space E has the Dunford-Pettis property (D.P.P.) if for every pair of sequences(x n ) in E and (x ′n ) in E ′, both weakly convergent to zero, we have that (x′n (x n )) tends to zero. P. Cembranos [Bull. Austral. Math. Soc. 28 (1983), no. 2, 175–186;] has proved that, if K is a compact Hausdorff dispersed space, then the following holds: For every Banach E with the D.P.P., the Banach space C(K,E) of continuous functions from K into E has the D.P.P.
In the note under review the author proves that this property characterizes compact dispersed spaces.
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P. CEMBRANOS: «On Banach spaces of vector valued continuous functions», Bull. Aust.Math. Soc., 28, 175-186 (1983).
J.DIESTEL: «Vector measures»,American Math. Soc.,Providence, R.I. (1977).
J.DIESTEL:«A survey of results related to the Dunford-Pettis property»,en «Proceedings of the conference on Integration,Topology and Geometry in Linear spaces»,Amer. Math. Soc., Providence, R. I. (1979).
A. GROTHENDIECK: «Sur les aplications linéaires faiblement compactes d'espaces du type C(K)», Cañad. J. Math., 5, 129-173 (1953).
H. E. LACEY: The isometric theory of classical Banach spaces, Springer, Berlin (1974).
M. TALAGRAND: «La propriété de Dunford-Pettis dans C(K, E) et Ü (E)», Israel J. of Math., 44, 317-321 (1983).