Implicit Function Theorem. Part I

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In this article, we formalize in Mizar [1], [3] the existence and uniqueness part of the implicit function theorem. In the first section, some composition properties of Lipschitz continuous linear function are discussed. In the second section, a definition of closed ball and theorems of several properties of open and closed sets in Banach space are described. In the last section, we formalized the existence and uniqueness of continuous implicit function in Banach space using Banach fixed point theorem. We referred to [7], [8], and [2] in this formalization.

[1] 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.

[2] Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003.

[3] Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015. doi: 10.1007/s10817-015-9345-1.

[4] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.

[5] 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.

[6] 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.

[7] Laurent Schwartz. Th´eorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.

[8] Laurent Schwartz. Calcul diff´erentiel, tome 2. Analyse. Hermann, 1997.

[9] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.

Formalized Mathematics

(a computer assisted approach)

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SCImago Journal Rank (SJR) 2017: 0.119
Source Normalized Impact per Paper (SNIP) 2017: 0.237

Target Group

researchers in the fields of formal methods and computer-checked mathematics


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