Ryll-Nardzewski fixed point theorem
- Ryll-Nardzewski fixed point theorem
In functional analysis, the Ryll-Nardzewski fixed point theorem states that if is a normed vector space and is a nonempty convex subset of which 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 a fixed point for each of the set's members.)
Applications
The Ryll-Nardzewski theorem yields the existence of a Haar measure on compact groups.
ee 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.
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