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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