In
functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if
is a
normed vector space and
is a nonempty
convex subset of
that is
compact under the
weak topology, then every
group (or equivalently: every
semigroup) of
affine isometries of
has at least one fixed point. (Here, a ''fixed point'' of a set of maps is a point that is
fixed by each map in the set.)
This theorem was announced by
Czesław Ryll-Nardzewski. Later Namioka and Asplund gave a proof based on a different approach. Ryll-Nardzewski himself gave a complete proof in the original spirit.
Applications
The Ryll-Nardzewski theorem yields the existence of a
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though ...
on compact groups.
See also
*
Fixed-point theorems
*
Fixed-point theorems in infinite-dimensional spaces In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations.
The first res ...
*
Markov-Kakutani fixed-point theorem - abelian semigroup of continuous affine self-maps on compact convex set in a topological vector space has a fixed point
References
* Andrzej Granas and
James Dugundji, ''Fixed Point Theory'' (2003) Springer-Verlag, New York, .
A proof written by J. Lurie
{{Functional analysis
Fixed-point theorems
Theorems in functional analysis