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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 is a nonempty convex closed subset of a Hausdorff locally convex topological vector space and is a continuous mapping of into itself such that is contained in a compact subset of , then has a fixed point.

History
The theorem was conjectured and proven for special cases, such as Banach spaces, by Juliusz Schauder in 1930. His conjecture for the general case was published in the Scottish book. In 1934, Tychonoff proved the theorem for the case when K is a compact convex subset of a locally convex space. This version is known as the Schauder–Tychonoff fixed-point theorem. B. V. Singbal proved the theorem for the more general case where K may be non-compact; the proof can be found in the appendix of Bonsall's book (see references). ==See also==
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