For a number of generalizations of the Brouwer fixed point theorem to infinite dimensions, see [[fixed point theorems in infinite-dimensional spaces]].
Generalizations of the Brouwer fixed point theorem to infinite dimensions include the [[Schauder fixed point theorem]] (if ''C'' is a [[nonempty]] [[closed set|closed]] [[convex]] subset of a [[Banach space]] and ''f'' is a continuous map from ''C'' to ''C'' whose image is [[compact|countably compact]], then ''f'' has a fixed point) and the [[Tychonoff fixed point theorem]] (if ''C'' is a nonempty compact convex subset of a [[locally convex]] [[topological vector space]], then any continuous map ''f'' from ''C'' to ''C'' has a fixed point).
