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On $ r$-separated sets in normed spaces


Author: Juan Arias-de-Reyna
Journal: Proc. Amer. Math. Soc. 112 (1991), 1087-1094
MSC: Primary 46B20; Secondary 47H09, 47H10
DOI: https://doi.org/10.1090/S0002-9939-1991-1059622-4
MathSciNet review: 1059622
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Abstract: The separation of a bounded set $ A$ in a metric space $ \delta (A)$ is defined as the supremum of the numbers $ r > 0$ such that there exists a sequence $ ({x_n})$ in $ A$ such that $ d({x_n},{x_m}) > r$ for every $ n \ne m$. We prove for every bounded set $ A$ in a Banach space that $ \delta (A) = \delta ({\text{co}}(A))$ where $ {\text{co}}(A)$ denotes the convex hull of $ A$. This yields a generalization of Darbo's fixed point theorem.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1059622-4
Article copyright: © Copyright 1991 American Mathematical Society

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