{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T02:21:54Z","timestamp":1768011714932,"version":"3.49.0"},"reference-count":9,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2017,12,20]],"date-time":"2017-12-20T00:00:00Z","timestamp":1513728000000},"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":[[2017,12,20]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p> 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.<\/jats:p>","DOI":"10.1515\/forma-2017-0026","type":"journal-article","created":{"date-parts":[[2018,3,28]],"date-time":"2018-03-28T22:16:22Z","timestamp":1522275382000},"page":"269-281","source":"Crossref","is-referenced-by-count":8,"title":["Implicit Function Theorem. Part I"],"prefix":"10.1515","volume":"25","author":[{"given":"Kazuhisa","family":"Nakasho","sequence":"first","affiliation":[{"name":"Osaka University, Osaka , Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuichi","family":"Futa","sequence":"additional","affiliation":[{"name":"Tokyo University of Technology Tokyo , Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[{"name":"Shinshu University, Nagano , Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2018,3,28]]},"reference":[{"key":"2021040814230074784_j_forma-2017-0026_ref_001_w2aab3b7b2b1b6b1ab1ab1Aa","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Bylinski, Adam Grabowski, Artur Korni\u0142owicz, 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 17.10.1007\/978-3-319-20615-817","DOI":"10.1007\/978-3-319-20615-8"},{"key":"2021040814230074784_j_forma-2017-0026_ref_002_w2aab3b7b2b1b6b1ab1ab2Aa","unstructured":"[2] Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003."},{"key":"2021040814230074784_j_forma-2017-0026_ref_003_w2aab3b7b2b1b6b1ab1ab3Aa","unstructured":"[3] Adam Grabowski, Artur Korni\u0142owicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015. doi: 10.1007\/s10817-015-9345-1.10.1007\/s10817-015-9345-1"},{"key":"2021040814230074784_j_forma-2017-0026_ref_004_w2aab3b7b2b1b6b1ab1ab4Aa","unstructured":"[4] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004."},{"key":"2021040814230074784_j_forma-2017-0026_ref_005_w2aab3b7b2b1b6b1ab1ab5Aa","unstructured":"[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\/v10037-011-0009-2.10.2478\/v10037-011-0009-2"},{"key":"2021040814230074784_j_forma-2017-0026_ref_006_w2aab3b7b2b1b6b1ab1ab6Aa","unstructured":"[6] Hideki Sakurai, Hiroyuki Okazaki, and Yasunari Shidama. Banach\u2019s continuous inverse theorem and closed graph theorem. Formalized Mathematics, 20(4):271-274, 2012. doi: 10.2478\/v10037-012-0032-y.10.2478\/v10037-012-0032-y"},{"key":"2021040814230074784_j_forma-2017-0026_ref_007_w2aab3b7b2b1b6b1ab1ab7Aa","unstructured":"[7] Laurent Schwartz. Th\u00b4eorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997."},{"key":"2021040814230074784_j_forma-2017-0026_ref_008_w2aab3b7b2b1b6b1ab1ab8Aa","unstructured":"[8] Laurent Schwartz. Calcul diff\u00b4erentiel, tome 2. Analyse. Hermann, 1997."},{"key":"2021040814230074784_j_forma-2017-0026_ref_009_w2aab3b7b2b1b6b1ab1ab9Aa","unstructured":"[9] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/25\/4\/article-p269.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2017-0026","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,9]],"date-time":"2021-04-09T16:35:33Z","timestamp":1617986133000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2017-0026"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,20]]},"references-count":9,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,3,28]]},"published-print":{"date-parts":[[2017,12,20]]}},"alternative-id":["10.1515\/forma-2017-0026"],"URL":"https:\/\/doi.org\/10.1515\/forma-2017-0026","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2017,12,20]]}}}