Kakutani fixed-point theorem: Difference between revisions

Revision as of 13:51, 19 December 2023

Revision as of 13:52, 19 December 2023

Line 147:

== Relation to Brouwer's fixed-point theorem ==

{{Disputed section|Mistake in section "[[Kakutani_fixed-point_theorem#Relation_to_Brouwer's_fixed-point_theorem|Relation to Brouwer's fixed-point theorem]]"|date=December 2023}}

Brouwer's fixed-point theorem is a special case of Kakutani fixed-point theorem. Conversely, Kakutani fixed-point theorem is an immediate generalization via the [[Selection theorem|approximate selection theorem]]:<ref>{{Cite book |last=Shapiro |first=Joel H. |title=A Fixed-Point Farrago |date=2016 |publisher=Springer International Publishing |isbn=978-3-319-27978-7 |pages=68–70 |oclc=984777840}}</ref>

{{Math proof|proof=

By the approximate selection theorem{{dubious}}, there exists a sequence of continuous <math>f_n: S \to S</math> such that <math>\operatorname{graph}(f_n) \subset[\operatorname{graph}(\varphi)]_{1/n}</math>. By Brouwer fixed-point theorem, there exists a sequence <math>x_n</math> such that <math>f_n(x_n) = x_n</math>, so <math>(x_n, x_n) \in [\operatorname{graph}(\varphi)]_{1/n}</math>.

Since <math>S</math> is compact, we can take a convergent subsequence <math>x_n \to x</math>. Then <math>(x, x)\in \operatorname{graph}(\varphi)</math> since it is a closed set.