{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,28]],"date-time":"2026-05-28T04:06:56Z","timestamp":1779941216333,"version":"3.53.1"},"reference-count":13,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2018,10,1]],"date-time":"2018-10-01T00:00:00Z","timestamp":1538352000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,10,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In this article, using the Mizar system [1], [2], we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized.<\/jats:p>\n                  <jats:p>In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to [4], [13], and [3] in this formalization.<\/jats:p>","DOI":"10.2478\/forma-2018-0021","type":"journal-article","created":{"date-parts":[[2019,2,24]],"date-time":"2019-02-24T04:31:04Z","timestamp":1550982664000},"page":"231-237","source":"Crossref","is-referenced-by-count":5,"title":["Continuity of Bounded Linear Operators on Normed Linear Spaces"],"prefix":"10.2478","volume":"26","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"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[{"name":"Shinshu University , Nagano , Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2019,2,23]]},"reference":[{"key":"2026052803403460845_j_forma-2018-0021_ref_001_w2aab3b7b3b1b6b1ab1ab1Aa","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":"2026052803403460845_j_forma-2018-0021_ref_002_w2aab3b7b3b1b6b1ab1ab2Aa","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":"2026052803403460845_j_forma-2018-0021_ref_003_w2aab3b7b3b1b6b1ab1ab3Aa","unstructured":"[3] N. J. Dunford and T. Schwartz. Linear operators I. Interscience Publ., 1958."},{"key":"2026052803403460845_j_forma-2018-0021_ref_004_w2aab3b7b3b1b6b1ab1ab4Aa","unstructured":"[4] Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972."},{"key":"2026052803403460845_j_forma-2018-0021_ref_005_w2aab3b7b3b1b6b1ab1ab5Aa","doi-asserted-by":"crossref","unstructured":"[5] Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Topological properties of real normed space. 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