A fixed point theorem for the infinite-dimensional simplex

Details A fixed point theorem for the infinite-dimensional simplex A fixed point theorem for the infinite-dimensional simplex

2006-10-24

arxiv.org/abs/math/0610707v1

J. Math. Anal. Appl. 332 (2007) 1063-1070

Mathematics

General Topology

8 pages; related work at http://www.math.hmc.edu/~su/papers.html

Scientific paper

10.1016/j.jmaa.2006.10.077

We define the infinite dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^infinity, and prove that this space has the 'fixed point property': any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.

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