functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if

E

is a

normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...

and

K

is a nonempty

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

subset of

E

that is

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

under the

weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...

, then every

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

(or equivalently: every

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

) of

affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...

isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

K

has at least one fixed point. (Here, a ''fixed point'' of a set of maps is a point that is

fixed Fixed may refer to: * ''Fixed'' (EP), EP by Nine Inch Nails * ''Fixed'', an upcoming 2D adult animated film directed by Genndy Tartakovsky * Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System * ...

by each map in the set.) This theorem was announced by

Czesław Ryll-Nardzewski Czesław Ryll-Nardzewski (; 7 October 1926 – 18 September 2015) was a Polish mathematician. Born in Wilno, Second Polish Republic (now Vilnius, Lithuania), he was a student of Hugo Steinhaus. At the age of 26 he became professor at Warsaw Uni ...

. 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 on compact groups.

References

* Andrzej Granas and

James Dugundji James Dugundji (August 30, 1919 – January 8, 1985) was an American mathematician, a professor of mathematics at the University of Southern California.. See in particulap. 244for a brief biography of Dugundji.

, ''Fixed Point Theory'' (2003) Springer-Verlag, New York, .
A proof written by J. Lurie
{{Functional analysis Fixed-point theorems Theorems in functional analysis

Applications

See also

References