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Pole DC | Wartość | Język |
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dc.contributor.author | Nakasho, Kazuhisa | - |
dc.contributor.author | Futa, Yuichi | - |
dc.date.accessioned | 2021-08-30T07:50:15Z | - |
dc.date.available | 2021-08-30T07:50:15Z | - |
dc.date.issued | 2021 | - |
dc.identifier.citation | Formalized Mathematics, Volume 29, Issue 1, Pages 9-19 | pl |
dc.identifier.issn | 1426-2630 | - |
dc.identifier.uri | http://hdl.handle.net/11320/11408 | - |
dc.description.abstract | In this article we formalize in Mizar [1], [2] the inverse function theorem for the class of C1 functions between Banach spaces. In the first section, we prove several theorems about open sets in real norm space, which are needed in the proof of the inverse function theorem. In the next section, we define a function to exchange the order of a product of two normed spaces, namely 𝔼 ↶ ≂ (x, y) ∈ X × Y ↦ (y, x) ∈ Y × X, and formalized its bijective isometric property and several differentiation properties. This map is necessary to change the order of the arguments of a function when deriving the inverse function theorem from the implicit function theorem proved in [6]. In the third section, using the implicit function theorem, we prove a theorem that is a necessary component of the proof of the inverse function theorem. In the last section, we finally formalized an inverse function theorem for class of C1 functions between Banach spaces. We referred to [9], [10], and [3] in the formalization. | pl |
dc.description.sponsorship | This study has been supported in part by JSPS KAKENHI Grant Numbers JP20K19863 and JP17K00182. | pl |
dc.language.iso | en | pl |
dc.publisher | DeGruyter Open | pl |
dc.rights | Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) | - |
dc.rights.uri | https://creativecommons.org/licenses/by-sa/3.0/ | - |
dc.subject | inverse function theorem | pl |
dc.subject | Lipschitz continuit | pl |
dc.subject | differentiability | pl |
dc.subject | implicit function | pl |
dc.subject | inverse function | pl |
dc.title | Inverse Function Theorem. Part I | pl |
dc.type | Article | pl |
dc.rights.holder | © 2021 University of Białymstoku | pl |
dc.rights.holder | CC-BY-SA License ver. 3.0 or later | pl |
dc.identifier.doi | 10.2478/forma-2021-0002 | - |
dc.description.Affiliation | Kazuhisa Nakasho - Yamaguchi University, Yamaguchi, Japan | pl |
dc.description.Affiliation | Yuichi Futa - Tokyo University of Technology, Tokyo, Japan | pl |
dc.description.references | Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17. | pl |
dc.description.references | Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6. | pl |
dc.description.references | Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003. | pl |
dc.description.references | Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321–327, 2004. | pl |
dc.description.references | Kazuhisa Nakasho. Invertible operators on Banach spaces. Formalized Mathematics, 27(2):107–115, 2019. doi:10.2478/forma-2019-0012. | pl |
dc.description.references | Kazuhisa Nakasho and Yasunari Shidama. Implicit function theorem. Part II. Formalized Mathematics, 27(2):117–131, 2019. doi:10.2478/forma-2019-0013. | pl |
dc.description.references | Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Implicit function theorem. Part I. Formalized Mathematics, 25(4):269–281, 2017. doi:10.1515/forma-2017-0026. | pl |
dc.description.references | Hideki Sakurai, Hiroyuki Okazaki, and Yasunari Shidama. Banach’s continuous inverse theorem and closed graph theorem. Formalized Mathematics, 20(4):271–274, 2012. doi:10.2478/v10037-012-0032-y. | pl |
dc.description.references | Laurent Schwartz. Th´eorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997. | pl |
dc.description.references | Laurent Schwartz. Calcul diff´erentiel, tome 2. Analyse. Hermann, 1997. | pl |
dc.identifier.eissn | 1898-9934 | - |
dc.description.volume | 29 | pl |
dc.description.issue | 1 | pl |
dc.description.firstpage | 9 | pl |
dc.description.lastpage | 19 | pl |
dc.identifier.citation2 | Formalized Mathematics | pl |
dc.identifier.orcid | 0000-0003-1110-4342 | - |
Występuje w kolekcji(ach): | Formalized Mathematics, 2021, Volume 29, Issue 1 |
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