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*It can be used to give a proof to the [[Nash embedding theorem]].<ref>{{cite journal |first=Matthias|last=Günther|title=Zum Einbettungssatz von J. Nash | trans-title=On the embedding theorem of J. Nash | language=de | journal=[[Mathematische Nachrichten]]|volume= 144 |year=1989|pages= 165–187|doi=10.1002/mana.19891440113 | mr=1037168}}</ref>
*It can be used to prove existence and uniqueness of solutions to value iteration, policy iteration, and policy evaluation of [[reinforcement learning]].<ref>{{cite book |first=Frank L. |last=Lewis |first2=Draguna |last2=Vrabie |first3=Vassilis L. |last3=Syrmos |title=Optimal Control |chapter=Reinforcement Learning and Optimal Adaptive Control |___location=New York |publisher=John Wiley & Sons |year=2012 |isbn=978-1-118-12272-3 |pages=461–517 [p. 474] |chapter-url=https://www.google.com/books/edition/_/U3Gtlot_hYEC?hl=en&gbpv=1&pg=PA474 }}</ref>
*It can be used to prove existence and uniqueness of an equilibrium in [[Cournot competition]],<ref>{{cite journal |first=Ngo Van |last=Long |first2=Antoine |last2=Soubeyran |title=Existence and Uniqueness of Cournot Equilibrium: A Contraction Mapping Approach |journal=[[Economics Letters]] |volume=67 |issue=3 |year=2000 |pages=345–348 |doi=10.1016/S0165-1765(00)00211-1 |url=https://www.cirano.qc.ca/pdf/publication/99s-22.pdf }}</ref> and other dynamic economic models.<ref>{{cite book |first=Nancy L. |last=Stokey|author1-link=Nancy Stokey |first2=Robert E. Jr. |last2=Lucas |author-link2=Robert Lucas Jr. |title=Recursive Methods in Economic Dynamics |___location=Cambridge |publisher=Harvard University Press |year=1989 |isbn=0-674-75096-9 |pages=508–516 |url=https://www.google.com/books/edition/_/BgQ3AwAAQBAJ?hl=en&gbpv=1&pg=PA508 }}</ref>
==Converses==
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