Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T05:59:20.576Z Has data issue: false hasContentIssue false

Separable measures and the Dunford-Pettis property

Published online by Cambridge University Press:  17 April 2009

José Aguayo
Affiliation:
Departamento de Matemática, Universidad de Concepción, Casilla 3-C. Concepción, Chile
José Sánchez
Affiliation:
Departamento de Matemática, Universidad de Concepción, Casilla 3-C. Concepción, Chile
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a complete regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremumnorm.

In this paper we give a characterisation of weakly compact operators defined from Cb(X) into a Banach space E which are β-continuous, where β is a locally convex topology on Cb(X) introduced by Wheeler. We also prove that (Cb(X), β) has the strict Dunford-Pettis property and, if X is a σ-compact space, (Cb(X), β), has the Dunford-Pettis property.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Aguayo, J. and Sanchez, J., ‘Weakly compact operators and the strict topologies’, Bull. Austral. Math. Soc. 39 (1989), 353359.Google Scholar
[2]Grothendieck, A., ‘Sur les applications lineaires faiblement compactes d'espaces du type C(K)’, Canad. J. Math. 5 (1953), 129173.Google Scholar
[3]Khurana, S., ‘Dunford-Pettis property’, J. Math. Anal. Appl. 65 (1978), 361364.Google Scholar
[4]Khurana, S., ‘Topologies on spaces of vector valued continuous functions’, Trans. Amer. Math. Soc. 241 (1978), 195211.CrossRefGoogle Scholar
[5]Koumoullis, G., ‘Perfect μ-additive measures and strict topologies’, Illinois J. of Math. 26 (1982), 466478.CrossRefGoogle Scholar
[6]Pachl, J., ‘Free uniform measures’, Comment. Math. Univ. Carolin. 15 (1974), 541553.Google Scholar
[7]Schaeffer, A., Topological vector spaces (Macmillan, New York) (Collier-Macmillan, London 1966).Google Scholar
[8]Sentilles, F., ‘Bounded countinuous functions on completely regular spaces’, Trans. Amer. Math. Soc. 168 (1972), 311336.Google Scholar
[9]Sentilles, F. and Wheeler, R., ‘Linear Functionals and partition of the unity in C b(X)’, Duke Math. J. 41 (1974), 483496.CrossRefGoogle Scholar
[10]Varadarajan, V., ‘Measures on topological spaces’, Amer. Math. Soc. Transl. (2) 48 (1965), 161220.Google Scholar
[11]Wheeler, R., ‘The strict topology, separable measure, and paracompactness’, Pacific J. Math. 47 (1973), 287302.CrossRefGoogle Scholar