A Fixed Point Theorem for the Infinite-Dimensional Simplex

Authors

Douglas Rizzolo '08, Harvey Mudd CollegeFollow
Francis E. Su, Harvey Mudd CollegeFollow

Student Co-author

HMC Undergraduate

Document Type

Article - preprint

Department

Mathematics (HMC)

Publication Date

8-2007

Abstract

We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^∞, 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.

Comments

Author's pre-print manuscript available for download.

For the publisher's version, please visit http://dx.doi.org/10.1016/j.jmaa.2006.10.077.

Rights Information

© 2007 Elsevier

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

DOI

10.1016/j.jmaa.2006.10.077

Recommended Citation

Douglas Rizzolo and Francis Edward Su. A fixed point theorem for the infinite-dimensional simplex. J. Math. Anal. Appl., 332(2):1063–1070, 2007.