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Archimedes integrals and conuclear spaces

Part of: Measures, integration, derivative, holomorphy Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

B. Jefferies  and
S. Okada
B. Jefferies
Affiliation:
Department of Mathematics, The University of Wollongong, Wollongong, N.S.W. 2500, Australia
S. Okada
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, W.A. 6150, Australia
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Abstract

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The lack of completeness with respect to the semivariation norm, of the space of Banach space valued functions, Pettis integrable with respect to a measure μ, often impedes the direct extension of results involving integral representations, true in the finite-dimensional setting, to the general vector space setting. It is shown here that the space of functions with values in a space Y, μ-Archimedes integrable in a Banach space X embedded in Y, is complete with respect to convergence in semivariation, provided the embedding from X into Y is completely summing. The result is applied to the case when Y is a conuclear space, in particular, when X is a function space continuously included in a space of distributions.

Keywords

Archimedes integralcompletenessconvergence in meanabsolutely summingconuclear space

MSC classification

Secondary: 46G10: Vector-valued measures and integration 47B10: Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 1 , August 1989 , pp. 22 - 31
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Bourbaki, N., Espaces vectorielles topologiques (Éléments de Mathématique, Livre V, Hermann, Paris, 1966).Google Scholar
[2]Diestel, J. and Uhl, J. J. Jr, Vector measures (Math Surveys 15, Amer. Math. Soc., Providence, R. I., 1977).Google Scholar
[3]Kelley, J. L. and Srinivasan, T. P., ‘On the Bochner integral’, Vector and operator valued measures and applications, pp. 165–174 (Academic Press, New York, 1973).Google Scholar
[4]Okada, S., ‘Integration of vector valued functions’, Measure theory and its applications, pp. 247–257 (Lecture Notes in Math., vol. 1033, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983).Google Scholar
[5]Okada, S. and Okazaki, Y., ‘Injective absolutely summing operators’, Math. Proc. Cambridge Philos. Soc., to appear.Google Scholar
[6]Pietsch, A., Nuclear locally convex spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar
[7]Schwartz, L., Radon measures in arbitrary topological spaces and cylindrical measures (Oxford University Press, London, 1973).Google Scholar
[8]Thomas, G. E. F., The Lebesgue-Nikodym theorem for vector-valued Radon measures, Mem. Amer. Math. Soc. 139 (1974).Google Scholar