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dc.contributor.authorNakasho, Kazuhisa-
dc.contributor.authorOkazaki, Hiroyuki-
dc.contributor.authorShidama, Yasunari-
dc.identifier.citationFormalized Mathematics, Volume 29, Issue 3, Pages 117-127pl
dc.descriptionThis study was supported in part by JSPS KAKENHI Grant Numbers 17K00182 and
dc.description.abstractIn 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 TOP-REAL(n), REAL-NS(n), and n-VectSp over F Real. We referred to [4], [9], and [8] in the
dc.publisherDeGruyter Openpl
dc.subjectreal vector spacepl
dc.subjecttopological spacepl
dc.subjectnormed spacespl
dc.titleReal Vector Space and Related Notionspl
dc.rights.holder© 2021 University of Białymstokupl
dc.rights.holderCC-BY-SA License ver. 3.0 or laterpl
dc.description.AffiliationKazuhisa Nakasho - Yamaguchi University, Yamaguchi, Japanpl
dc.description.AffiliationHiroyuki Okazaki - Shinshu University, Nagano, Japanpl
dc.description.AffiliationYasunari Shidama - Shinshu University, Nagano, Japanpl
dc.description.referencesGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, 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/
dc.description.referencesGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/
dc.description.referencesNoboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577–580,
dc.description.referencesMiyadera Isao. Functional Analysis. Riko-Gaku-Sya,
dc.description.referencesRobert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339–345,
dc.description.referencesKarol Pąk. Basic properties of the rank of matrices over a field. Formalized Mathematics, 15(4):199–211, 2007. doi:10.2478/
dc.description.referencesKarol Pąk. Linear map of matrices. Formalized Mathematics, 16(3):269–275, 2008. doi:10.2478/
dc.description.referencesLaurent Schwartz. Theorie des ensembles et topologie, tome 1. Analyse. Hermann,
dc.description.referencesKosaku Yosida. Functional Analysis. Springer,
dc.identifier.citation2Formalized Mathematicspl
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