{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T13:54:16Z","timestamp":1777470856137,"version":"3.51.4"},"reference-count":9,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2021,9,1]],"date-time":"2021-09-01T00:00:00Z","timestamp":1630454400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,9,1]]},"abstract":"<jats:p>\n                    <jats:bold>Summary<\/jats:bold>\n                    . In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. In the Mizar language [1], [2], variables are strictly typed, and their type conversion requires a complicated process. Our purpose is to formalize that some properties of finite dimensional vector spaces are preserved in type transformations, and to contain the complexity of type transformations into this paper. Specifically, we show that properties such as algebraic structure, subsets, finite sequences and their sums, linear combination, linear independence, and affine independence are preserved in type conversions among\n                    <jats:monospace>TOP-REAL(n)<\/jats:monospace>\n                    ,\n                    <jats:monospace>REAL-NS(n)<\/jats:monospace>\n                    , and\n                    <jats:monospace>n-VectSp over F Real<\/jats:monospace>\n                    . We referred to [4], [9], and [8] in the formalization.\n                  <\/jats:p>","DOI":"10.2478\/forma-2021-0012","type":"journal-article","created":{"date-parts":[[2022,1,6]],"date-time":"2022-01-06T01:56:02Z","timestamp":1641434162000},"page":"117-127","source":"Crossref","is-referenced-by-count":2,"title":["Real Vector Space and Related Notions"],"prefix":"10.2478","volume":"29","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":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hiroyuki","family":"Okazaki","sequence":"additional","affiliation":[{"name":"Shinshu University Nagano , 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":[[2021,12,30]]},"reference":[{"key":"2026042801401589202_j_forma-2021-0012_ref_1","doi-asserted-by":"crossref","unstructured":"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":"2026042801401589202_j_forma-2021-0012_ref_2","doi-asserted-by":"crossref","unstructured":"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":"2026042801401589202_j_forma-2021-0012_ref_3","unstructured":"Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577\u2013580, 2005."},{"key":"2026042801401589202_j_forma-2021-0012_ref_4","unstructured":"Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972."},{"key":"2026042801401589202_j_forma-2021-0012_ref_5","unstructured":"Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339\u2013345, 1996."},{"key":"2026042801401589202_j_forma-2021-0012_ref_6","doi-asserted-by":"crossref","unstructured":"Karol P\u0105k. Basic properties of the rank of matrices over a field. Formalized Mathematics, 15(4):199\u2013211, 2007. doi:10.2478\/v10037-007-0024-5.10.2478\/v10037-007-0024-5","DOI":"10.2478\/v10037-007-0024-5"},{"key":"2026042801401589202_j_forma-2021-0012_ref_7","doi-asserted-by":"crossref","unstructured":"Karol P\u0105k. Linear map of matrices. Formalized Mathematics, 16(3):269\u2013275, 2008. doi:10.2478\/v10037-008-0032-0.10.2478\/v10037-008-0032-0","DOI":"10.2478\/v10037-008-0032-0"},{"key":"2026042801401589202_j_forma-2021-0012_ref_8","unstructured":"Laurent Schwartz. Th\u00e9orie des ensembles et topologie, tome 1. Analyse. Hermann, 1997."},{"key":"2026042801401589202_j_forma-2021-0012_ref_9","unstructured":"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-2021-0012","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T13:40:33Z","timestamp":1777383633000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2021-0012"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,1]]},"references-count":9,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,12,30]]},"published-print":{"date-parts":[[2021,9,1]]}},"alternative-id":["10.2478\/forma-2021-0012"],"URL":"https:\/\/doi.org\/10.2478\/forma-2021-0012","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,1]]}}}