Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-16T21:33:53.107Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Amemiya, I., Okada, S. and Okazaki, Y.. Pre-Radon measures on topological spaces. Kodai Math. J. 1 (1978), 101–132.CrossRefGoogle Scholar
[2]Comfort, W. W. and Negrepontis, S.. Chain Conditions in Topology (Cambridge University Press, 1982).CrossRefGoogle Scholar
[3]Dancer, N. and Sims, B.. Weak Star separability. Bull. Austral. Math. Soc. 20 (1979), 253–257.CrossRefGoogle Scholar
[4]Diestel, J.. Sequences and Series in Banach Spaces (Springer-Verlag, 1984).CrossRefGoogle Scholar
[5]Diestel, J. and Uhl, J. J. Jr. Vector Measures (American Mathematical Society, 1977).CrossRefGoogle Scholar
[6]Jefferies, B. and Okada, S.. Archimedes integrals and conuclear spaces, preprint.Google Scholar
[7]Köthe, G.. Topological Vector Spaces 1 (English translation) (Springer-Verlag, 1969).Google Scholar
[8]Negrepontis, S.. Banach spaces and topology. In Handbook of Set-theoretic Topology (North Holland, 1984), pp. 1045–1142.CrossRefGoogle Scholar
[9]Okada, S.. Integration of vector valued functions. In Measure Theory and its Applications, Lecture Notes in Math, vol 1033 (Springer-Verlag, 1983).CrossRefGoogle Scholar
[10]Pietsch, A.. Absolut p-summierende Abbildungen in normierten Räumen. Studia Math. 28 (1967), 333–353.CrossRefGoogle Scholar
[11]Pietsch, A.. Nuclear Locally Convex Spaces (English translation) (Springer-Verlag, 1972).Google Scholar
[12]Rosenthal, H.. On injective Banach spaces and the spaces ℒ∞(μ) for finite measures μ. Acta Math. 124 (1970), 205–248.CrossRefGoogle Scholar
[13]Schwartz, L.. Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures (Oxford University Press, 1973).Google Scholar
[14]Thomas, G. E. F.. The Lebesgue–Nikodym theorem for vector-valued Radon measures. Mem. Amer. Math. Soc. 139 (1974).Google Scholar