Ryll-Nardzewski fixed-point theorem

The fixed point theorem of Ryll - Nardzewski, named after Czesław Ryll - Nardzewski, is a set from the mathematical branch of functional analysis. The theorem ensures the existence of a common fixed point of a family of certain images of a compact convex set in itself.

Wording of the sentence

Let be a locally convex space, for example a normed space, and be a non-empty weakly compact convex set. Next a non-empty family of mappings with the following properties is:

Then there is at least one common fixed point of, that is to say: There is a such that for all.

Comments

  • The proof is, first, that every finite subset of has a fixed point, and then closes with a compactness argument on the assertion.
  • The requirement that non- contractive should be, is automatically satisfied if all the elements of isometries of a normed space are. This special case is also called the fixed point theorem of Ryll - Nardzewski: Any semigroup weakly continuous affine isometries of weakly compact convex set into itself has a fixed point.

Application

The best-known application is the derivation of the existence of the hair - measure on a compact group. The space of finite Borel measures on is on the dual space of the space of continuous functions, and therefore carries the weak -* topology, which makes it a locally convex space whose weak topology is precisely this weak -* topology. As a convex set is taken. For and were explained by the formulas. Define further by

Then a semigroup of isometries that into itself. If one applies to this situation, the fixed point theorem of Ryll - Nardzewski on, we obtain a measure that can be easily detected as a hair - measure.

Swell

  • John B. Conway: A Course in Functional Analysis, Springer- Verlag ( 1994), ISBN 0387972455
  • C. Ryll - Nardzewski: On fixed points of semi- groups of endomorphisms of linear spaces, Proc. Fifth Berkeley Sympos. Math Statist. and Probability, Univ. California Press, Berkeley (1967 ), pp. 55-61
  • Functional Analysis
  • Set ( mathematics)
Affine transformation Isometry
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