Completeness and the contraction principle

Borwein, J. M.

doi:10.1090/S0002-9939-1983-0681829-1

Completeness and the contraction principle
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Proc. Amer. Math. Soc. 87 (1983), 246-250 Request permission

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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