×

On integration in Banach spaces. VI. (English) Zbl 0628.28007

The authors continue the investigations concerning a variant of integration of Banach space valued functions [see the previous parts of this paper (Part V: ibid. 30(105), 610-628 (1980; Zbl 0506.28004)], introduced by the first author. They prove here a mean value theorem for measures verifying the axiom of Price. As applications, they obtain a certain Lagrange-type theorem for Fréchet differentiability in normed spaces, extending some classical results (including one of L. Schwartz). The method of proof uses S-integrability of Komogorov, which is supplementary analyzed in the last paragraph of the paper.
Reviewer: Gr.Arsene

MSC:

28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration

Citations:

Zbl 0506.28004
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] Bartle R.: A general bilinear integral. Studia Math. 15 (1956), 337-352. · Zbl 0070.28102
[2] [2J Birkhoff G.: Integration of functions with values in a Banach space. Trans. Amer. Math. Soc. 38 (1935), 357-378. · Zbl 0013.00803
[3] Bochner S.: Integration von Funktionen deren Werte die Element eines Vectorraumes sind. Fund. Math. 20 (1933), 262-276. · Zbl 0007.10901
[4] Bryant W. W.: Independent axioms for convexity. J. Geometry 5 (1974), 95-99. · Zbl 0288.52001
[5] Bryant W. W.: Topological convexity spaces. Proc. Edinburgh Math. Soc. 19 (1974), 125-132. · Zbl 0292.46007
[6] Bryant W. W., Webster R. J.: Convexity spaces I. J. Math. Anal. Appl. 37 (1972), 206-213. · Zbl 0197.48401
[7] Bryant W. W., Webster R. J.: Convexity spaces II. J. Math. Anal. Appl. 43 (1973), 321-327. · Zbl 0259.52001
[8] Bryant W. W., Webster R. J.: Convexity spaces III. J. Math. Anal. Appl. 57 (1977), 382 to 392. · Zbl 0342.52001
[9] Danzer L., Grünbaum B., Klee V.: Helly’s theorems and its relatives. Proc. Sympos. Pure Math. Vol. VII(convexity). Amer. Math. Soc. (1963), 101-180. · Zbl 0132.17401
[10] Diestel J., Uhl J. J.: Vector measures. Math. Surveys 15, Amer. Math. Soc., Providence, Rhode Island 1977. · Zbl 0369.46039
[11] Dieudonné J.: Foundations of Modern Analysis. Academic Press, New York 1960. · Zbl 0100.04201
[12] Dinculeanu N.: Vector measures. Pergamon Press, New York 1967. · Zbl 0142.10502
[13] Dobrakov I.: On integration in Banach spaces I. Czechoslovak Math. J. 20 (1970), 511- 536. · Zbl 0215.20103
[14] Dobrakov I.: On integration in Banach spaces II. Czechoslovak Math. J. 20 (1970), 680- 695. · Zbl 0224.46050
[15] Dobrakov I.: On integration in Banach spaces III. Czechoslovak Math. J. 29 (1979), 478-499. · Zbl 0429.28011
[16] Dobrakov I.: On integration, in Banach spaces IV. Czechoslovak Math. J. 30 (1980), 259-279. · Zbl 0452.28006
[17] Dobrakov I.: On integration in Banach spaces V. Czechoslovak Math. J. 30 (1980), 610- 622. · Zbl 0506.28004
[18] Dunford N.: Integration in general analysis. Trans. Amer. Math. Soc. 37 (1935), 441 - 453. · Zbl 0013.15502
[19] Dunford N., Schwartz J. T.: Linear operators. Part I. Interscience Publishers, Inc., New York 1964. · Zbl 0128.34803
[20] Eckhoff J.: Der Satz von Radon in konvexen Produktstructuren I. Monatsh. Math. 72 (1968), 303-314. · Zbl 0159.51701
[21] Eckhoff J.: Der Satz von Radon in konvexen Produktstructuren II. Monatsh. Math. 73 (1969), 7-30. · Zbl 0174.53701
[22] Ellis J. W.: A general set-separation theorem. Duke Math. J. 19 (1952), All-All. · Zbl 0047.28601
[23] Green J. W., Gustin W.: Quasiconvex sets. Canad. J. Math. 2 (1950), 489-507. · Zbl 0038.35501
[24] Guay M. D., Naimpally S. A.: Characterization of a convex subspace of a linear topological space. Math. Japonicae 20 (1975), 37-41. · Zbl 0335.46008
[25] Hanner O.: Connectedness and convex hulls. Seminar on Convex Sets, Inst. for Advanced Study, Princeton, New Jersey (1949-1950), 35-40.
[26] Hildebrandt T. H.: On bounded linear functional operations. Trans. Amer. Math. Soc. 36 (1934), 868-875. · Zbl 0010.30303
[27] Kay D. C, Womble E. W.: Axiomatic convexity theory and relationships between the Carathéorody, Helly and Radon numbers. Pacific J. Math. 38 (1971), 471 - 485. · Zbl 0235.52001
[28] Kelley J. L., Srinivasan T. P.: On the Bochner integral. Proc. Conf. Vector and Operator-valued Measures, Snowbird, Utah, Academic Press (1973), 165-174. · Zbl 0293.28010
[29] Kolmogoroff A.: Untersuchunge über den Integralbegriff. Math. Ann. 103 (1930), 654-696. · JFM 56.0923.01
[30] Levi F. W.: On Helly’s theorem and the axioms of convexity. J. Indian Math. Soc. (N.S.) 15(1951), 65-76. · Zbl 0044.19101
[31] Mah P., Naimpally S. A., Whitfield J. H. M.: Linearization of a convexity space. J. London Math. Soc. (2) 13 (1976), 209-214. · Zbl 0326.52005
[32] Maynard H. B.: A general Radon-Nikodym theorem. Proc. Conf. Vector and Operator-valued Measures, Snowbird, Utah, Academic Press (1973), 233 - 246. · Zbl 0301.28006
[33] McLeod R.: Mean value theorem for vector valued functions. Proc. Edinburgh Math. Soc. 14(1965), 197-209. · Zbl 0135.34301
[34] Morales P.: Mean value theorem for the Dobrakov integral. Proc. Conf. on Measure Theory and Appl., Northern Illinois University (1981), 235-242. · Zbl 0523.28009
[35] Prenowitz W.: A contemporary approach to classical geometry. H.E. Slaught Memorial Paper 9, Amer. Math. Monthly 68 (1961), Appendix, 1–67. · Zbl 0094.15402
[36] Prenowitz W., Jantosciak J.: Geometries and join spaces. J. Reine Angew. Math. 257 (1972), 100-128. · Zbl 0264.50002
[37] Price G. B.: The theory of integration. Trans. Amer. Math. Soc. 47 (1940), 1 - 50. · Zbl 0022.31901
[38] Reay J. R.: Carathéodory theorems in convex product structures. Pacific J. Math. 35 (1970), 227-230. · Zbl 0182.55601
[39] Schwartz L.: Un lemme sur la dérivation des fonctions vectorielles d’une variable réelle. Ann. Inst. Fourier 2 (1950), 17-18. · Zbl 0042.11601
[40] Schwartz L.: Espaces de fonctions différentiables à valeurs vectorielles. J. Analyse Math. 4 (1956), 88-148. · Zbl 0066.09601
[41] Schwartz L.: Analyse mathématique I. Hermann, Paris 1967. · Zbl 0382.52002
[42] Szafron S. A., Weston J. H.: An internal solution to the problem of linearization of a convexity space. Canad. Math. Bull. 19 (1976), 487-494. · Zbl 0382.52002
[43] Taylor A. E.: Introduction to functional analysis. John Wiley and Sons, Inc., New York 1958. · Zbl 0081.10202
[44] Yamamuro S.: Differential calculus in topological linear spaces. Lecture Notes in Math. 374, Springer-Verlag, New York 1974. · Zbl 0276.58001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.