A fixed point theorem for the infinite-dimensional simplex
General Topology
2007-08-28 v1 Classical Analysis and ODEs
Combinatorics
Abstract
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.
Cite
@article{arxiv.math/0610707,
title = {A fixed point theorem for the infinite-dimensional simplex},
author = {Douglas Rizzolo and Francis Edward Su},
journal= {arXiv preprint arXiv:math/0610707},
year = {2007}
}
Comments
8 pages; related work at http://www.math.hmc.edu/~su/papers.html