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Proceedings of the American Mathematical Society

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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
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Abstract: If $ ({x_n})$ is a bounded sequence in a Banach space, is there an element $ x = \sum\nolimits_{n = 1}^\infty {{a_n}{x_n}} $ sucn that $ \sum\nolimits_{n = 1}^\infty {\left\Vert {{a_n}{x_n}} \right\Vert < \infty } $ and tne directional derivative of the norm at $ x$, $ D(x,{x_n})$, exists for every $ n$? In fact, there are such $ x$'s dense in the closed span of $ \left\{ {{x_n}} \right\}$. An application of this fact is made to a proof of Rybakov's theorem on vector measures.


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  • [1] 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
  • [2] 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
  • [3] 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
  • [4] S. Mazur, Über konvexe mengen in linearen normierte raumen, Studia Math. 4 (1933), 70-84.
  • [5] V. I. Rybakov, On the theorem of Bartle, Dunford and Schwartz on vector-valued measures, Mat. Zametki 7 (1970), 247–254 (Russian). MR 0260971

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0774002-1
Keywords: Continuous convex function, Gateaux differentiable, countably additive measure
Article copyright: © Copyright 1985 American Mathematical Society