Abstract Bochner and McShane integrals
Abstract
A short description of the classical Bochner integral is presented together with the McShane concept of integration based on Riemann type integral sums. The corresponding classes are compared and it will be shown that the situation is different for finite- and infinite- dimensional valued vector functions.
References
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2. N. Dunford, J.T. Schwartz, Linear operators I., Interscience Publishers, New York, London, 1958.
3. A. Dvoretzky, C.A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. (USA) 36 (1950), 192-197.
4. D.H. Fremlin, J. Mendoza, The integration of vector valued functions, Illinois J. of Math. 38 (1994), 127-147.
5. R. Gordon, The McShane Integral of Banach - valued functions, Illinois J. of Math. 34 (1990), 557-567.
6. R. Gordon, Riemann integration in Banach spaces, Rocky Mountain J. of Math. 21 (1991), 923-949.
7. R. Henstock, Lectures on the Theory of Integration, World Scientific, Singapore, 1988.
8. R. Henstock, The General Theory of Integration, Clarendon Press, Oxford, 1991.
9. E. Hille, R.S. Phillips, Functional Analysis and Semi-groups, American Mathematical Society, Providence, 1957.
10. Y. Kubota, A note on McShane's integral, Math. Japonica 30 (1985), 57-62.
11. J. Kurzweil, Nichtabsolut konvergente Integrate, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980.
12. S. Lang, Real and Functional Analysis, Springer - Verlag, New York, 1993.
13. J. Mawhin, Analyse. Fondements, techniques, évolution, De Boeck - Wesmael, Bruxelles, 1992.
14. E.J. McShane, Unified Integration, Academic Press, New York, London, 1983.
15. J. Mendoza, On Lebesgue integrability of McShane integrable functions, Real Analysis Exchange 18 (2) (1992/93), 456-458.
16. Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Singapore, 1992.
17. K. Yosida, Functional Analysis, Academic Press, New York, 1965.
Schwabik Štefan. (1996). Abstract Bochner and McShane integrals. Annales Mathematicae Silesianae, 10, 21-56. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14191
Štefan Schwabik
SCHWABIK@EARN.CVUT.CZ
Matematyický ústav AV ČR, Czech Republik Czechia
Matematyický ústav AV ČR, Czech Republik Czechia
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