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 formula_1 is a nonempty convex closed subset of a Hausdorff topological vector space formula_2 and formula_3 is a continuous mapping of formula_1 into itself such that formula_5 is contained in a compact subset of formula_1, then formula_3 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 proved earlier by Juliusz Schauder and Jean Leray.
The statement is as follows:
Let formula_3 be a continuous and compact mapping of a Banach space formula_9 into itself, such that the set
is bounded. Then formula_3 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).