{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T11:24:25Z","timestamp":1777461865164,"version":"3.51.4"},"reference-count":11,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"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":[[2019,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In this article, various definitions of contuity of multilinear operators on normed linear spaces are discussed in the Mizar formalism [4], [1] and [2]. In the first chapter, several basic theorems are prepared to handle the norm of the multilinear operator, and then it is formalized that the linear space of bounded multilinear operators is a complete Banach space.<\/jats:p>\n                  <jats:p>In the last chapter, the continuity of the multilinear operator on finite normed spaces is addressed. Especially, it is formalized that the continuity at the origin can be extended to the continuity at every point in its whole domain. We referred to [5], [11], [8], [9] in this formalization.<\/jats:p>","DOI":"10.2478\/forma-2019-0006","type":"journal-article","created":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T05:33:30Z","timestamp":1558071210000},"page":"61-65","source":"Crossref","is-referenced-by-count":0,"title":["Continuity of Multilinear Operator on Normed Linear Spaces"],"prefix":"10.2478","volume":"27","author":[{"given":"Kazuhisa","family":"Nakasho","sequence":"first","affiliation":[{"name":"Yamaguchi University , Yamaguchi , 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":[[2019,5,16]]},"reference":[{"key":"2026042801383709004_j_forma-2019-0006_ref_001_w2aab3b7b5b1b6b1ab1ab1Aa","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":"2026042801383709004_j_forma-2019-0006_ref_002_w2aab3b7b5b1b6b1ab1ab2Aa","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":"2026042801383709004_j_forma-2019-0006_ref_003_w2aab3b7b5b1b6b1ab1ab3Aa","doi-asserted-by":"crossref","unstructured":"[3] Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. The product space of real normed spaces and its properties. Formalized Mathematics, 15(3):81\u201385, 2007. doi:10.2478\/v10037-007-0010-y.10.2478\/v10037-007-0010-y","DOI":"10.2478\/v10037-007-0010-y"},{"key":"2026042801383709004_j_forma-2019-0006_ref_004_w2aab3b7b5b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Adam Grabowski, Artur Korni\u0142owicz, and Adam Naumowicz. Four decades of Mizar. 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Formalized Mathematics, 12(3):269\u2013275, 2004."},{"key":"2026042801383709004_j_forma-2019-0006_ref_008_w2aab3b7b5b1b6b1ab1ab8Aa","unstructured":"[8] Laurent Schwartz. Th\u00e9orie des ensembles et topologie, tome 1. Analyse. Hermann, 1997."},{"key":"2026042801383709004_j_forma-2019-0006_ref_009_w2aab3b7b5b1b6b1ab1ab9Aa","unstructured":"[9] Laurent Schwartz. Calcul diff\u00e9rentiel, tome 2. Analyse. Hermann, 1997."},{"key":"2026042801383709004_j_forma-2019-0006_ref_010_w2aab3b7b5b1b6b1ab1ac10Aa","unstructured":"[10] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39\u201348, 2004."},{"key":"2026042801383709004_j_forma-2019-0006_ref_011_w2aab3b7b5b1b6b1ab1ac11Aa","unstructured":"[11] Kosaku Yoshida. Functional Analysis. Springer, 1980."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/27\/1\/article-p61.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2019-0006","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T13:15:47Z","timestamp":1777382147000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2019-0006"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,1]]},"references-count":11,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,5,16]]},"published-print":{"date-parts":[[2019,4,1]]}},"alternative-id":["10.2478\/forma-2019-0006"],"URL":"https:\/\/doi.org\/10.2478\/forma-2019-0006","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2019,4,1]]}}}