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Kakutani's Fixed Point Theorem


Kakutani's fixed point theorem is a result in functional analysis which establishes the existence of a common fixed point among a collection of maps defined on certain "well-behaved" subsets of locally convex topological vector spaces. The theorem is relevant both because of its independent theoretical significance and because of other results which stem as corollaries therefrom.

One common form of Kakutani's fixed point theorem states that, given a locally convex topological vector space X, any equicontinuous group G={g_alpha}_(alpha in A) of affine maps g_alpha mapping a (nonempty) compact, convex subset K subset X into itself necessarily has a common fixed point in K, i.e., that under the above conditions, there exists a point x in K satisfying g_alpha(x)=x for all g_alpha in G. In addition, Markov and Kakutani proved that some of these hypotheses can be weakened without affecting the results, e.g., that the result remains true for arbitrary topological vector spaces X (which may or may not be locally convex) provided that G is a commuting collection (not necessarily a group, and not necessarily equicontinuous) of continuous affine maps from K into K.

Kakutani's fixed point theorem is an important ingredient in a number of other results, notable among which is the proof that, on every compact topological group G, there exists a translation-invariant Haar measure.


See also

Affine Transformation, Equicontinuous, Fixed Point, Fixed Point Theorem, Haar Measure

Portions of this entry contributed by Christopher Stover

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References

Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.

Referenced on Wolfram|Alpha

Kakutani's Fixed Point Theorem

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Kakutani's Fixed Point Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KakutanisFixedPointTheorem.html

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