Schauder fixed-point theorem

The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to locally convex topological vector spaces, which may be of infinite dimension. It asserts that if K {\displaystyle K} is a nonempty convex closed subset of a Hausdorff locally convex topological vector space V {\displaystyle V} and f {\displaystyle f} is a continuous mapping of K {\displaystyle K} into itself such that f ( K ) {\displaystyle f(K)} is contained in a compact subset of K {\displaystyle K} , then f {\displaystyle f} has a fixed point.

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