A Fixed Point Theorem for the Infinite-Dimensional Simplex

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.