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Injective absolutely summing operators

Published online by Cambridge University Press:  24 October 2008

Susumu Okada  and
Yoshiaki Okazaki
Susumu Okada
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, Murdoch, W.A. 6150, Australia
Yoshiaki Okazaki
Affiliation:
Department of Mathematics, Kyushu University, 33, Fukuoka 812, Japan
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Extract

Let X be an infinite-dimensional Banach space. It is well-known that the space of X-valued, Pettis integrable functions is not always complete with respect to the topology of convergence in mean, that is, the uniform convergence of indefinite integrals (see [14]). The Archimedes integral introduced in [9] does not suffer from this defect. For the Archimedes integral, functions to be integrated are allowed to take values in a locally convex space Y larger than the space X while X accommodates the values of indefinite integrals. Moreover, there exists a locally convex space Y, into which X is continuously embedded, such that the space ℒ(μX, Y) of Y-valued, Archimedes integrable functions is identical to the completion of the space of X valued, simple functions with repect to the toplogy of convergence in mean, for each non-negative measure μ (see [9]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 3 , May 1988 , pp. 497 - 502
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

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