Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T22:40:43.635Z Has data issue: false hasContentIssue false
  • English
  • Français

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
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.

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