Every continuous map from a nonempty compact convex set in a Banach space to itself has a fixed point. A useful equivalent form takes a nonempty closed bounded convex set and a continuous map whose image is relatively compact. To obtain this version from the compact-domain version, restrict to and use closed convex hull of a compact set.
For a proof on , the Schauder approximation lemma gives continuous maps with finite-dimensional and . The Brouwer fixed-point theorem on the nonempty compact convex set gives , so . A convergent subsequence as and the continuity of give an exact fixed point.
Let be a nonempty compact convex set in a Banach space and a continuous map. For every , there is a finite-dimensional linear subspace and a continuous map that approximates uniformly to within in norm.
Choose whose open radius- balls cover the compact image . Set andThe denominator is positive. This convex combination belongs to and depends continuously on . Every contributing has distance less than from , so the triangle inequality gives .
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The Schauder fixed-point theorem is a fundamental result in fixed-point theory, particularly in the context of functional analysis and topology. It provides conditions under which a continuous function mapping a convex compact subset of a Banach space (or more generally, in a topological vector space) has at least one fixed point.