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Keywords:
Henstock-Kurzweil integral; McShane integral
Summary:
This note contains a simple example which does clearly indicate the differences in the Henstock-Kurzweil, McShane and strong McShane integrals for Banach space valued functions.
References:
[1] M. Day: Normed Linear Spaces. Springer, New York, 1962. Zbl 0100.10802
[2] D. H. Fremlin: The Henstock and McShane integrals of vector-valued functions. Illinois J. Math. 38 (1994), 471–479. MR 1269699 | Zbl 0797.28006
[3] D. H.  Fremlin, J. Mendoza: On the integration of vector-valued functions. Illinois J. Math. 38 (1994), 127–147. MR 1245838
[4] R. Gordon: The Integrals of Lebesgue, Denjoy, Perron and Henstock. Amer. Math. Soc., Providence, 1991. MR 1288751
[5] R. Gordon: The McShane integral of Banach-valued functions. Illinois J. Math. 34 (1990), 557–567. MR 1053562 | Zbl 0685.28003
[6] E. Hille, R. Phillips: Functional Analysis and Semigroups. Amer. Math. Soc., Providence, 1957. MR 0089373
[7] Peng Yee Lee, R. Výborný: The Integral: an Easy Approach after Kurzweil and Henstock. Cambridge Univ. Press, Cambridge, 2000. MR 1756319
[8] E. J. McShane: Unified Integration. Academic Press, New York, 1983. MR 0740710 | Zbl 0551.28001
[9] W. Pfeffer: The Riemann Approach to Integration. Cambridge Univ. Press, Cambridge, 1993. MR 1268404 | Zbl 0804.26005
[10] Š. Schwabik: Abstract Bochner and McShane integrals. Ann. Math. Silesianae 10 (1996), 21–56. MR 1399609 | Zbl 0868.28005
[11] Š. Schwabik, Guoju Ye: On the strong McShane integral of functions with values in a Banach space. Czechoslovak Math. J. 51 (2001), 819–828. MR 1864044
[12] C. Swartz: Beppo Levi’s theorem for the vector-valued McShane integral and applications. Bull. Belgian Math. Soc. 4 (1997), 589–599. MR 1600292 | Zbl 1038.46505
[13] C. Swartz: Infinite Matrices and the Gliding Hump. World Sci. Publ., Singapore, 1996. MR 1423136 | Zbl 0923.46003
[14] C. Swartz: Introduction to Gauge Integrals. World Scientific Publ., Singapore, 2001. MR 1845270 | Zbl 0982.26006
[15] Wu Congin, Yao Xiaobo: A Riemann-type definition of the Bochner integral. J. Math. Study 27 (1994), 32–36. MR 1318255
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