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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Strongly extreme points and the Radon-Nikodým property


Author: Zhibao Hu
Journal: Proc. Amer. Math. Soc. 118 (1993), 1167-1171
MSC: Primary 46B22
DOI: https://doi.org/10.1090/S0002-9939-1993-1152279-5
MathSciNet review: 1152279
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Abstract: We prove that if $ K$ is a bounded and convex subset of a Banach space $ X$ and $ x$ is a point in $ K$, then $ x$ is a strongly extreme point of $ K$ if and only if $ x$ is a strongly extreme point of $ {\overline K ^{\ast}}$ which is the weak$ ^{{\ast}}$ closure of $ K$ in $ {X^{{\ast}{\ast}}}$. We also prove that a Banach space $ X$ has the Radon-Nikodým property if and only if for any equivalent norm on $ X$, the unit ball has a strongly extreme point.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1152279-5
Keywords: Radon-Nikodým property, extreme point, strongly extreme point
Article copyright: © Copyright 1993 American Mathematical Society