Schauder fixed-point theorem

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

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