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Completeness and the contraction principle


Author: J. M. Borwein
Journal: Proc. Amer. Math. Soc. 87 (1983), 246-250
MSC: Primary 54H25; Secondary 54E40
DOI: https://doi.org/10.1090/S0002-9939-1983-0681829-1
MathSciNet review: 681829
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Abstract: We prove (something more general than) the result that a convex subset of a Banach space is closed if and only if every contraction of the space leaving the convex set invariant has a fixed point in that subset. This implies that a normed space is complete if and only if every contraction on the space has a fixed point. We also show that these results fail if "convex" is replaced by "Lipschitz-connected" or "starshaped".


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DOI: https://doi.org/10.1090/S0002-9939-1983-0681829-1
Keywords: Contraction mapping, complete metric space, Ekeland's principle
Article copyright: © Copyright 1983 American Mathematical Society

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