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SOME COMMENTS ON SCALAR DIFFERENTIATION OF VECTOR-VALUED FUNCTIONS

Part of: Real functions Normed linear spaces and Banach spaces; Banach lattices Measures, integration, derivative, holomorphy

Published online by Cambridge University Press:  11 November 2014

K. M. NARALENKOV
K. M. NARALENKOV*
Affiliation:
Moscow State Institute of International Relations, Department of Mathematical Methods and Information Technologies, Vernadskogo Ave. 76, 119454 Moscow, Russian Federation email [email protected]
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Abstract

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We make some comments on the existence, uniqueness and integrability of the scalar derivatives and approximate scalar derivatives of vector-valued functions. We are particularly interested in the connection between scalar differentiation and the weak Radon–Nikodým property.

Keywords

scalar and approximate scalar derivativessVB and sVBG functionsRadon–Nikodým and weak Radon–Nikodým propertiesLipschitz functionindefinite Riemann integral

MSC classification

Primary: 46G05: Derivatives
Secondary: 26A45: Functions of bounded variation, generalizations 46B22: Radon-Nikodým, Kreĭn-Milman and related properties 46G10: Vector-valued measures and integration
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 2 , April 2015 , pp. 311 - 321
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Benyamini, Y. and Lindenstrauss, J., Geometric Nonlinear Functional Analysis, Vol. 1, American Mathematical Society Colloquium Publications, 48 (American Mathematical Society, Providence, RI, 2000).Google Scholar
Bongiorno, B., Di Piazza, L. and Musiał, K., ‘A variational Henstock integral characterization of the Radon–Nikodým property’, Illinois J. Math. 53(1) (2009), 87–99.Google Scholar
Bongiorno, B., Di Piazza, L. and Musiał, K., ‘A characterization of the weak Radon–Nikodým property by finitely additive interval functions’, Bull. Aust. Math. Soc. 80(3) (2009), 476–485.Google Scholar
Bourgain, J. and Rosenthal, H. P., ‘Martingales valued in certain subspaces of L 1’, Israel J. Math. 37(1–2) (1980), 54–75.Google Scholar
Bourgin, R. D., Geometric Aspects of Convex Sets with the Radon–Nikodým Property, Lecture Notes in Mathematics, 993 (Springer, Berlin, 1983).Google Scholar
Diestel, J. and Uhl, J. J. Jr, Vector Measures, Mathematical Surveys, 15 (American Mathematical Society, Providence, RI, 1977).Google Scholar
Diestel, J. and Uhl, J. J. Jr, ‘Progress in vector measures—1977–83’, in: Measure Theory and its Applications (Sherbrooke, Que., 1982), Lecture Notes in Mathematics, 1033 (Springer, Berlin, 1983), 144–192.Google Scholar
Dilworth, S. J. and Girardi, M., ‘Nowhere weak differentiability of the Pettis integral’, Quaest. Math. 18(4) (1995), 365–380.Google Scholar
Dunford, N. and Morse, A. P., ‘Remarks on the preceding paper of James A. Clarkson’, Trans. Amer. Math. Soc. 40(3) (1936), 415–420.Google Scholar
Gordon, R. A., ‘Integration and differentiation in a Banach space’, PhD Thesis, University of Illinois at Urbana-Champaign, 1987.Google Scholar
Gordon, R. A., ‘Differentiation in Banach spaces’, Preprint.Google Scholar
Gordon, R. A., ‘The Denjoy extension of the Bochner, Pettis, and Dunford integrals’, Studia Math. 92(1) (1989), 73–91.Google Scholar
Gordon, R. A., The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, 4 (American Mathematical Society, Providence, RI, 1994).CrossRefGoogle Scholar
Hájek, P., Montesinos Santalucía, V., Vanderwerff, J. and Zizler, V., Biorthogonal Systems in Banach Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 26 (Springer, New York, 2008).Google Scholar
Musiał, K., ‘The weak Radon–Nikodým property in Banach spaces’, Studia Math. 64(2) (1979), 151–173.CrossRefGoogle Scholar
Musiał, K., ‘Topics in the theory of Pettis integration’, Rend. Istit. Mat. Univ. Trieste 23(1) (1991), 177–262.Google Scholar
Myung, P. J., ‘The Denjoy extension of the Riemann and McShane integrals’, Czechoslovak Math. J. 50(3) (2000), 615–625.Google Scholar
Naralenkov, K. M., ‘Asymptotic structure of Banach spaces and Riemann integration’, Real Anal. Exchange 33(1) (2007/08), 111–124.Google Scholar
Naralenkov, K., ‘On Denjoy type extensions of the Pettis integral’, Czechoslovak Math. J. 60(3) (2010), 737–750.Google Scholar
Pettis, B. J., ‘On integration in vector spaces’, Trans. Amer. Math. Soc. 44(2) (1938), 277–304.CrossRefGoogle Scholar
Pettis, B. J., ‘Differentiation in Banach spaces’, Duke Math. J. 5(2) (1939), 254–269.Google Scholar
Rodríguez, J., ‘The Bourgain property and convex hulls’, Math. Nachr. 280(11) (2007), 1302–1309.Google Scholar
Solomon, D. W., ‘On differentiability of vector-valued functions of a real variable’, Studia Math. 29 (1967), 1–4.CrossRefGoogle Scholar
Stefánsson, G. F., ‘Pettis integrability’, Trans. Amer. Math. Soc. 330(1) (1992), 401–418.Google Scholar
Songwei, Q., ‘Nowhere differentiable Lipschitz maps and the Radon–Nikodým property’, J. Math. Anal. Appl. 185(3) (1994), 613–616.Google Scholar
Talagrand, M., ‘Pettis integral and measure theory’, Mem. Amer. Math. Soc. 51(307) (1984).Google Scholar
Thomson, B. S., ‘Characterizations of an indefinite Riemann integral’, Real Anal. Exchange 35(2) (2010), 487–492.Google Scholar