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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The sum of two Radon-Nikodým-sets need not be a Radon-Nikodým-set


Author: Walter Schachermayer
Journal: Proc. Amer. Math. Soc. 95 (1985), 51-57
MSC: Primary 46B22; Secondary 28B05, 46G10
MathSciNet review: 796445
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Abstract: It was shown by C. Stegall that, if $ C$ is a Radon-Nikodym-set and $ K$ weakly compact, then $ K + C$ is a Radon-Nikodym-set. We show that there are closed, bounded, convex Radon-Nikodym-sets $ {C_1}$ and $ {C_2}$ such that $ {C_1} + {C_2}$ is closed but contains an isometric copy of the unit-ball of $ {c_0}$. In fact, we give two examples, one following the lines of one due to McCartney and O'Brian, the other due to Bourgain and Delbaen. We also give an easy example of a non-Radon-Nikodym-set $ C$ such that, for every $ \varepsilon > 0$, there is a Radon-Nikodym-set $ {C_\varepsilon }$ such that $ C$ is contained in the sum of $ {C_\varepsilon }$ and the ball of radius $ \varepsilon $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0796445-2
PII: S 0002-9939(1985)0796445-2
Article copyright: © Copyright 1985 American Mathematical Society