Implicit Function Theorem. Part II

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In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here.

In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.

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  • [6] 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.

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

  • [8] 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.

  • [9] 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.

  • [10] Laurent Schwartz. Théorie des ensembles et topologie tome 1. Analyse. Hermann 1997.

  • [11] Laurent Schwartz. Calcul différentiel tome 2. Analyse. Hermann 1997.

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