Editing Schauder fixed-point theorem (section)

{{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]].)