Brouwer Fixed-point Theorem - Statement

Statement

The theorem has several formulations, depending on the context in which it is used. The simplest is sometimes given as follows:

In the plane

Every continuous function f from a closed disk to itself has at least one fixed point.

This can be generalized to an arbitrary finite dimension:

In Euclidean space

Every continuous function from a closed ball of a Euclidean space to itself has a fixed point.

A slightly more general version is as follows:

Convex compact set

Every continuous function f from a convex compact subset K of a Euclidean space to K itself has a fixed point.

An even more general form is better known under a different name:

Schauder fixed point theorem

Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point.