{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:07:55Z","timestamp":1783901275337,"version":"3.55.0"},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In this article we formalize in Mizar [1], [2] the inverse function theorem for the class of\n                    <jats:italic>C<\/jats:italic>\n                    <jats:sup>1<\/jats:sup>\n                    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 \ud835\udd3c \u21b6 \u2242 (\n                    <jats:italic>x, y<\/jats:italic>\n                    ) \u2208\n                    <jats:italic>X \u00d7 Y<\/jats:italic>\n                    \u21a6 (\n                    <jats:italic>y, x<\/jats:italic>\n                    ) \u2208\n                    <jats:italic>Y \u00d7 X<\/jats:italic>\n                    , 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].\n                  <\/jats:p>\n                  <jats:p>\n                    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\n                    <jats:italic>C<\/jats:italic>\n                    <jats:sup>1<\/jats:sup>\n                    functions between Banach spaces. We referred to [9], [10], and [3] in the formalization.\n                  <\/jats:p>","DOI":"10.2478\/forma-2021-0002","type":"journal-article","created":{"date-parts":[[2021,8,27]],"date-time":"2021-08-27T11:49:05Z","timestamp":1630064945000},"page":"9-19","source":"Crossref","is-referenced-by-count":1,"title":["Inverse Function Theorem. Part I\n                    <sup>1<\/sup>"],"prefix":"10.2478","volume":"29","author":[{"given":"Kazuhisa","family":"Nakasho","sequence":"first","affiliation":[{"name":"Yamaguchi University , Yamaguchi , Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuichi","family":"Futa","sequence":"additional","affiliation":[{"name":"Tokyo University of Technology Tokyo , Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2021,8,26]]},"reference":[{"key":"2026071212491126818_j_forma-2021-0002_ref_001","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, 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\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.10.1007\/978-3-319-20615-8_17","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026071212491126818_j_forma-2021-0002_ref_002","doi-asserted-by":"crossref","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.10.1007\/s10817-017-9440-6604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026071212491126818_j_forma-2021-0002_ref_003","unstructured":"[3] Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003."},{"key":"2026071212491126818_j_forma-2021-0002_ref_004","unstructured":"[4] Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321\u2013327, 2004."},{"key":"2026071212491126818_j_forma-2021-0002_ref_005","doi-asserted-by":"crossref","unstructured":"[5] Kazuhisa Nakasho. Invertible operators on Banach spaces. Formalized Mathematics, 27 (2):107\u2013115, 2019. doi:10.2478\/forma-2019-0012.10.2478\/forma-2019-0012","DOI":"10.2478\/forma-2019-0012"},{"key":"2026071212491126818_j_forma-2021-0002_ref_006","doi-asserted-by":"crossref","unstructured":"[6] Kazuhisa Nakasho and Yasunari Shidama. Implicit function theorem. Part II. Formalized Mathematics, 27(2):117\u2013131, 2019. doi:10.2478\/forma-2019-0013.10.2478\/forma-2019-0013","DOI":"10.2478\/forma-2019-0013"},{"key":"2026071212491126818_j_forma-2021-0002_ref_007","doi-asserted-by":"crossref","unstructured":"[7] Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Implicit function theorem. Part I. Formalized Mathematics, 25(4):269\u2013281, 2017. doi:10.1515\/forma-2017-0026.10.1515\/forma-2017-0026","DOI":"10.1515\/forma-2017-0026"},{"key":"2026071212491126818_j_forma-2021-0002_ref_008","doi-asserted-by":"crossref","unstructured":"[8] Hideki Sakurai, Hiroyuki Okazaki, and Yasunari Shidama. Banach\u2019s continuous inverse theorem and closed graph theorem. Formalized Mathematics, 20(4):271\u2013274, 2012. doi:10.2478\/v10037-012-0032-y.10.2478\/v10037-012-0032-y","DOI":"10.2478\/v10037-012-0032-y"},{"key":"2026071212491126818_j_forma-2021-0002_ref_009","unstructured":"[9] Laurent Schwartz. Th\u00e9orie des ensembles et topologie, tome 1. Analyse. Hermann, 1997."},{"key":"2026071212491126818_j_forma-2021-0002_ref_010","unstructured":"[10] Laurent Schwartz. Calcul diff\u00e9rentiel, tome 2. Analyse. Hermann, 1997."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2021-0002","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T23:17:03Z","timestamp":1783898223000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2021-0002"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,1]]},"references-count":10,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,8,26]]},"published-print":{"date-parts":[[2021,4,1]]}},"alternative-id":["10.2478\/forma-2021-0002"],"URL":"https:\/\/doi.org\/10.2478\/forma-2021-0002","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,1]]}}}