{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T01:13:45Z","timestamp":1783905225008,"version":"3.55.0"},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2022,7,1]],"date-time":"2022-07-01T00:00:00Z","timestamp":1656633600000},"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":[[2022,7,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>This paper, using the Mizar system [1], [2], provides useful tools for working with real linear spaces and real normed spaces. These include the identification of a real number set with a one-dimensional real normed space, the relationships between real linear spaces and real Euclidean spaces, the transformation from a real linear space to a real vector space, and the properties of basis and dimensions of real linear spaces. We referred to [6], [10], [8], [9] in this formalization.<\/jats:p>","DOI":"10.2478\/forma-2022-0008","type":"journal-article","created":{"date-parts":[[2022,12,24]],"date-time":"2022-12-24T05:16:56Z","timestamp":1671859016000},"page":"93-98","source":"Crossref","is-referenced-by-count":0,"title":["Transformation Tools for Real Linear Spaces"],"prefix":"10.2478","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1110-4342","authenticated-orcid":false,"given":"Kazuhisa","family":"Nakasho","sequence":"first","affiliation":[{"name":"Yamaguchi University , Yamaguchi , Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2022,12,24]]},"reference":[{"key":"2026071212490730455_j_forma-2022-0008_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.","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026071212490730455_j_forma-2022-0008_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\/s10817-017-9440-6.604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026071212490730455_j_forma-2022-0008_ref_003","unstructured":"[3] Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577\u2013580, 2005."},{"key":"2026071212490730455_j_forma-2022-0008_ref_004","doi-asserted-by":"crossref","unstructured":"[4] 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.","DOI":"10.2478\/v10037-007-0010-y"},{"key":"2026071212490730455_j_forma-2022-0008_ref_005","doi-asserted-by":"crossref","unstructured":"[5] Takao Inou\u00e9, Adam Naumowicz, Noboru Endou, and Yasunari Shidama. Partial differentiation, differentiation and continuity on n-dimensional real normed linear spaces. Formalized Mathematics, 19(2):65\u201368, 2011. doi:10.2478\/v10037-011-0011-8.","DOI":"10.2478\/v10037-011-0011-8"},{"key":"2026071212490730455_j_forma-2022-0008_ref_006","unstructured":"[6] Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972."},{"key":"2026071212490730455_j_forma-2022-0008_ref_007","doi-asserted-by":"crossref","unstructured":"[7] Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51\u201359, 2011. doi:10.2478\/v10037-011-0009-2.","DOI":"10.2478\/v10037-011-0009-2"},{"key":"2026071212490730455_j_forma-2022-0008_ref_008","unstructured":"[8] Laurent Schwartz. Th\u00e9orie des ensembles et topologie, tome 1. Analyse. Hermann, 1997."},{"key":"2026071212490730455_j_forma-2022-0008_ref_009","unstructured":"[9] Laurent Schwartz. Calcul diff\u00e9rentiel, tome 2. Analyse. Hermann, 1997."},{"key":"2026071212490730455_j_forma-2022-0008_ref_010","unstructured":"[10] K\u00f4saku Yosida. Functional Analysis. Springer, 1980."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2022-0008","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:45:49Z","timestamp":1783903549000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2022-0008"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,1]]},"references-count":10,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,12,24]]},"published-print":{"date-parts":[[2022,7,1]]}},"alternative-id":["10.2478\/forma-2022-0008"],"URL":"https:\/\/doi.org\/10.2478\/forma-2022-0008","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,1]]}}}