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{{Short description|Extension of the Brouwer fixed-point theorem}}
The '''Schauder fixed-point theorem''' is an extension of the [[Brouwer fixed-point theorem]] to [[locally convex topological vector space]]s, which may be of infinite dimension. It asserts that if <math>K</math> is a nonempty [[convex set|convex]] closed subset of a [[Hausdorff space|Hausdorff]] locally convex topological vector space <math>V</math> and <math>f</math> is a continuous mapping of <math>K</math> into itself such that <math>f(K)</math> is contained in a [[compact set|compact]] subset of <math>K</math>, then <math>f</math> has a [[fixed point (mathematics)|fixed point]].

A consequence, called '''[[Helmut Schaefer|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 <math>f</math> be a continuous and compact mapping of a Banach space <math>X</math> into itself, such that the set

: <math>
\{ x \in X : x = \lambda f(x) \mbox{ for some } 0 \leq \lambda \leq 1 \}
</math>

is bounded.  Then <math>f</math> has a fixed point. (A ''compact mapping'' in this context is one for which the image of every bounded set is [[relatively compact]].)

==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 Café|Scottish book]]. In 1934, [[Andrey Nikolayevich Tychonoff|Tychonoff]] proved the theorem for the case when ''K'' is a compact convex subset of a [[Locally convex topological vector space|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==
*  [[Fixed-point theorem]]s
*  [[Banach fixed-point theorem]]
*  [[Kakutani fixed-point theorem]]

==References==
* F. F. Bonsall, ''Lectures on some fixed point theorems of functional analysis'', Bombay 1962
* D. Gilbarg, [[N. Trudinger]], ''Elliptic Partial Differential Equations of Second Order''. {{ISBN|3-540-41160-7}}.
* H. Schaefer, ''Über die Methode der a priori-Schranken'', Math. Ann. 129, 415–416 (1955), {{doi|10.1007/BF01362380}}
* J. Schauder, ''Der Fixpunktsatz in Funktionalräumen'', Studia Math. 2 (1930), 171–180
* A. Tychonoff, ''Ein Fixpunktsatz'', Mathematische Annalen 111 (1935), 767–776
* E. Zeidler, ''Nonlinear Functional Analysis and its Applications, ''I'' - Fixed-Point Theorems''

==External links==
* {{springer|title=Schauder theorem|id=p/s083320}}
* {{planetmath reference|urlname=SchauderFixedPointTheorem|title=Schauder fixed point theorem}}  
* {{planetmath reference|urlname=ProofOfSchauderFixedPointTheorem|title=proof of Schauder Fixed Point Theorem}}.

{{Functional analysis}}
{{Topological vector spaces}}

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[[Category:Fixed-point theorems]]
[[Category:Theorems in functional analysis]]
[[Category:Topological vector spaces]]