Gâteaux differentiable points with special representation
Author: Seung Jae Oh
Journal: Proc. Amer. Math. Soc. 93 (1985), 456-458
MSC: Primary 46G05; Secondary 46A99, 46G10
MathSciNet review: 774002
Full-text PDF Free Access
Abstract: If is a bounded sequence in a Banach space, is there an element sucn that and tne directional derivative of the norm at , , exists for every ? In fact, there are such 's dense in the closed span of . An application of this fact is made to a proof of Rybakov's theorem on vector measures.
-  Russell G. Bilyeu and Paul W. Lewis, Orthogonality and the Hewitt-Yosida theorem in spaces of measures, Rocky Mountain J. Math. 7 (1977), no. 4, 629–638. MR 0450499, https://doi.org/10.1216/RMJ-1977-7-4-629
-  J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964
-  John R. Giles, Convex analysis with application in the differentiation of convex functions, Research Notes in Mathematics, vol. 58, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 650456
-  S. Mazur, Über konvexe mengen in linearen normierte raumen, Studia Math. 4 (1933), 70-84.
-  V. I. Rybakov, On the theorem of Bartle, Dunford and Schwartz on vector-valued measures, Mat. Zametki 7 (1970), 247–254 (Russian). MR 0260971
- R. G. Bilyeu and P. W. Lewis, Orthogonality and Hewitt-Yosida theorem in spaces of measures, Rocky Mountain J. Math. 7 (1977), 629-638. MR 0450499 (56:8793)
- J. Diestel and J. J. Uhl, Jr., Vector measures, Amer. Math. Soc., Providence, R. I., 1970. MR 0453964 (56:12216)
- J. R. Giles, Convex analysis with application in differentiation of convex functions, Pitman, Boston. Mass., 1982. MR 650456 (83g:46001)
- S. Mazur, Über konvexe mengen in linearen normierte raumen, Studia Math. 4 (1933), 70-84.
- V. Rybakov, Theorem of Bartle, Dunford, and Schwartz on vector- valued measures, Mat. Zametki 7 (1970), 247-254. MR 0260971 (41:5591)
Keywords: Continuous convex function, Gateaux differentiable, countably additive measure
Article copyright: © Copyright 1985 American Mathematical Society