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 is a convex subset of a topological vector space and is a continuous mapping of into itself so that is contained in a compact subset of, then has a fixed point.
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 Juliusz Schauder and Jean Leray. The statement is as follows:
Let be a continuous and compact mapping of a Banach space into itself, such that the set
is bounded. Then has a fixed point.
Read more about Schauder Fixed Point Theorem: History
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