Ryll-Nardzewski fixed point theorem

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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.)

[edit] Applications

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

[edit] See also

[edit] 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.