Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-25T03:54:42.001Z Has data issue: false hasContentIssue false
  • English
  • Français

Testing on null sequences is enough for Bochner integrability

Published online by Cambridge University Press:  17 April 2009

Fernando Mayoral  and
Pedro J. Paúl
Pedro J. Paúl
Affiliation:
E.S. Ingenieros Industriales, Avda. Reina Mercedes s/n, 41012–Sevilla, Spain email: PITI@CICA.ES
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 E be a normed space, a Fréchet space or a complete (DF)-space satisfying the dual density condition. Let Ω be a Radon measure space. We prove that a function f: Ω → Eis Bochner p-integrable if (and only if) fis p-integrable with respect to the topology of uniform convergence on the norm-null sequences from E′.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 54 , Issue 2 , October 1996 , pp. 299 - 307
Copyright
Copyright © Australian Mathematical Society 1996

References

REFERENCES

[1]Bierstedt, K.D. and Bonet, J., ‘Dual density conditions in (DF)-spaces. I’, Results Math. 14 (1988), 242–274.CrossRefGoogle Scholar
[2]Bourbaki, N., Intégration (Hermann, Paris, 1973).Google Scholar
[3]Diestel, J. and Uhl, J.J., Vector measures (American Mathematical Society, Providence, R.I., 1977).CrossRefGoogle Scholar
[4]Florencio, M., Mayoral, F. and Paúl, P.J., ‘Spaces of vector-valued integrable functions and localization of bounded subsets’, Math. Nachr. 174 (1995), 89–111.CrossRefGoogle Scholar
[5]Jarchow, H., Locally convex spaces (B.G. Teubner, Stuttgart, 1981).CrossRefGoogle Scholar
[6]Köthe, G., Topological vector spaces. I, II (Springer-Verlag, Berlin, Heidelberg, New York, 1969, 1979).Google Scholar
[7]Carreras, P. Pérez and Bonet, J., Barrelled locally convex spaces (North-Holland, Amsterdam, New York, Oxford, Tokyo, 1987).Google Scholar
[8]Robertson, A.P. and Robertson, W.J., Topological vector spaces, (second edition) (Oxford University Press, London, 1973).Google Scholar
[9]Schwartz, L., Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute of Fundamental Research Studies (North-Holland, Amsterdam, New York, Oxford, Tokyo, 1987).Google Scholar
[10]Thomas, G.E.F., ‘The Lebesgue-Nikodym theorem for vector valued Radon measures’, Mem. Amer. Math. Soc. 139 (1974).Google Scholar