Generalizations
The Brouwer fixed-point theorem forms the starting point of a number of more general fixed-point theorems.
The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitrary Hilbert space instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are not compact. For example, in the Hilbert space ℓ2 of square-summable real (or complex) sequences, consider the map f : ℓ2 → ℓ2 which sends a sequence (xn) from the closed unit ball of ℓ2 to the sequence (yn) defined by
It is not difficult to check that this map is continuous, has its image in the unit sphere of ℓ 2, but does not have a fixed point.
The generalizations of the Brouwer fixed-point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and in addition also often an assumption of convexity. See fixed-point theorems in infinite-dimensional spaces for a discussion of these theorems.
There is also finite-dimensional generalization to a larger class of spaces: If is a product of finitely many chainable continua, then every continuous function has a fixed point, where a chainable continuum is a (usually but in this case not necessarily metric) compact Hausdorff space of which every open cover has a finite open refinement, such that if and only if . Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers.
The Kakutani fixed point theorem generalizes the Brouwer fixed-point theorem in a different direction: it stays in Rn, but considers upper semi-continuous correspondences (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.
The Lefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of singular homology that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of D n.
Read more about this topic: Brouwer Fixed-point Theorem