Differentiability of convex functions and Rybakov's theorem
Authors:
Russell G. Bilyeu and Paul W. Lewis
Journal:
Proc. Amer. Math. Soc. 86 (1982), 186-187
MSC:
Primary 28B05; Secondary 46G10
DOI:
https://doi.org/10.1090/S0002-9939-1982-0663894-X
MathSciNet review:
663894
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Rybakov proved that if is a countably additive Banach space valued measure on a
-algebra
, then there is an element
so that
. In this note, we show that Rybakov's theorem follows essentially from a classical result of Mazur on the Gateaux differentiability of convex functions.
- [1] R. G. Bartle, N. Dunford and J. T. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289-305. MR 0070050 (16:1123c)
- [2] J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R. I., 1977. MR 0453964 (56:12216)
- [3] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
- [4] S. Mazur, Über konvexe Mengen in linearen normierte Räumen, Studia Math. 4 (1933), 70-84.
- [5] V. Rybakov, Theorem of Bartle, Dunford, and Schwartz on vector-valued measures, Mat. Zametki 7 (1970), 247-254. MR 0260971 (41:5591)
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28B05, 46G10
Retrieve articles in all journals with MSC: 28B05, 46G10
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1982-0663894-X
Article copyright:
© Copyright 1982
American Mathematical Society