Ryll-Nardzewski fixed-point theorem

In functional analysis, the Ryll-Nardzewski fixed point theorem states that if $E$ is a normed vector space and $K$ is a nonempty convex subset of $E$ which is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of $K$ has at least one fixed point. (Here, a fixed point of a set of maps is a point that is a fixed point for each of the set's members.)

Applications

The Ryll-Nardzewski theorem yields the existence of a Haar measure on (locally ?) compact groups.

See also

• Fixed point theorems
• Fixed point theorems in infinite-dimensional spaces

References

• C. Ryll-Nardzewski. On fixed points of semi-groups of endomorphisms of linear spaces. Proc. 5-th Berkeley Symp. Probab. Math. Stat., 2: 1, Univ. California Press (1967) pp. 55–61
• Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5.