Kakutani's fixed point theorem is a result in functional analysis which establishes the existence of a common fixed point among a collection of maps defined on certain "well-behaved" subsets of locally convex topological vector spaces. The theorem is relevant both because of its independent theoretical significance and because of other results which stem as corollaries therefrom.
One common form of Kakutani's fixed point theorem states that, given a locally convex topological vector space 
, any equicontinuous group 
of affine
maps 
mapping a (nonempty) compact, convex subset 
into itself necessarily has a common fixed
point in 
, i.e., that under the above conditions, there exists a point
satisfying 
for all 
.
In addition, Markov and Kakutani proved that some of these hypotheses can be weakened
without affecting the results, e.g., that the result remains true for arbitrary topological
vector spaces 
(which may or may not be locally convex) provided that 
is a commuting collection (not necessarily a group,
and not necessarily equicontinuous) of continuous
affine maps from 
into 
.
Kakutani's fixed point theorem is an important ingredient in a number of other results, notable among which is the proof that, on every compact topological
group 
,
there exists a translation-invariant Haar measure.
