<abstract xmlns="http://www.w3.org/1999/xhtml">

In this article we formalize in Mizar [1], [2] the inverse function theorem for the class of <italic>C</italic>1 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 ↶ ≂ (<italic>x, y</italic>) ∈ <italic>X × Y</italic> ↦ (<italic>y, x</italic>) ∈ <italic>Y × X</italic>, 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 <italic>C</italic>1 functions between Banach spaces. We referred to [9], [10], and [3] in the formalization.
</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.

Inverse Function Theorem. Part I1

Formalized Mathematics

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{"article-title":{"content":"Inverse Function Theorem. Part I","sup":1}}

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...

反函数定理。第一部分¹

在线发布:2021年8月26日

页面范围： 9 - 19

认可的:2021年3月30日

内政部： https://doi.org/10.2478/forma-2021-0002

关键词
反函数定理,利普希茨连续性,可微性,隐函数,逆函数

©2021 Kazuhisa Nakasho等人，由Sciendo出版

本作品根据Creative Commons Attribution-ShareAlike 4.0 International License获得许可。

反函数定理。第一部分1