On Implicit and Inverse Function Theorems on Euclidean Spaces

Nakasho, Kazuhisa; Shidama, Yasunari

doi:10.2478/forma-2022-0012

Abstract

Previous Mizar articles [7, 6, 5] formalized the implicit and inverse function theorems for Frechet continuously differentiable maps on Banach spaces. In this paper, using the Mizar system [1], [2], we formalize these theorems on Euclidean spaces by specializing them. We referred to [4], [12], [10], [11] in this formalization.

References

[1] Grzegorz Bancerek, Czesław Bylinacute;ski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, 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.
Open DOI Search in Google Scholar Back to article
[2] Grzegorz Bancerek, Czesław Bylinacute;ski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. 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/s10817-017-9440-6.10.1007/s10817-017-9440-6604425130069070
Search in Google Scholar Back to article
[3] Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces ℛn. Formalized Mathematics, 15(2):65–72, 2007. doi:10.2478/v10037-007-0008-5.
Open DOI Search in Google Scholar Back to article
[4] Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972.
Search in Google Scholar Back to article
[5] Kazuhisa Nakasho and Yuichi Futa. Inverse function theorem. Part I. Formalized Mathematics, 29(1):9–19, 2021. doi:10.2478/forma-2021-0002.
Open DOI Search in Google Scholar Back to article
[6] Kazuhisa Nakasho and Yasunari Shidama. Implicit function theorem. Part II. Formalized Mathematics, 27(2):117–131, 2019. doi:10.2478/forma-2019-0013.
Open DOI Search in Google Scholar Back to article
[7] 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.
Open DOI Search in Google Scholar Back to article
[8] Kazuhisa Nakasho, Hiroyuki Okazaki, and Yasunari Shidama. Real vector space and related notions. Formalized Mathematics, 29(3):117–127, 2021. doi:10.2478/forma-2021-0012.
Open DOI Search in Google Scholar Back to article
[9] Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.
Open DOI Search in Google Scholar Back to article
[10] Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.
Search in Google Scholar Back to article
[11] Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.
Search in Google Scholar Back to article
[12] Kôsaku Yosida. Functional Analysis. Springer, 1980.
Search in Google Scholar Back to article

On Implicit and Inverse Function Theorems on Euclidean Spaces

Abstract

Paradigm

My account