Schauder fixed point theorem

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The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to Banach spaces. It asserts that if $K$ is a compact, convex subset of a Banach space and $T$ is a mapping of $K$ into itself, then $T$ has a fixed point. It was discovered by Juliusz Schauder.

A consequence, called Schaefer's fixed point theorem, is particularly useful for proving existence of solutions to nonlinear partial differential equations. Schaefer's theorem is in fact a special case of the far reaching Leray-Schauder theorem which was discovered earlier by Schauder and Jean Leray. The statement is as follows. Let $T$ be a continuous and compact mapping of a Banach space $X$ into itself, such that the set

$\{ x \in X : x = \lambda T x \mbox{ for some } 0 \leq \lambda \leq 1 \}$