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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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Keywords: Continuous convex function, Gateaux differentiable, countably additive measure
Article copyright: © Copyright 1985 American Mathematical Society