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