{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T13:45:07Z","timestamp":1777470307556,"version":"3.51.4"},"reference-count":7,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"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":[[2021,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In this article, using the Mizar system [1], [2], first we give a definition of a functional space which is constructed from all continuous functions defined on a compact topological space [5]. We prove that this functional space is a Banach space [3]. Next, we give a definition of a function space which is constructed from all continuous functions with bounded support. We also prove that this function space is a normed space.<\/jats:p>","DOI":"10.2478\/forma-2021-0005","type":"journal-article","created":{"date-parts":[[2021,8,27]],"date-time":"2021-08-27T11:49:00Z","timestamp":1630064940000},"page":"49-62","source":"Crossref","is-referenced-by-count":1,"title":["Functional Space Consisted by Continuous Functions on Topological Space"],"prefix":"10.2478","volume":"29","author":[{"given":"Hiroshi","family":"Yamazaki","sequence":"first","affiliation":[{"name":"Nagano Prefectural Institute of Technology , Nagano , Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Keiichi","family":"Miyajima","sequence":"additional","affiliation":[{"name":"Ibaraki University , Ibaraki , Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[{"name":"Karuizawa Hotch 244-1 , Nagano , Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2021,8,26]]},"reference":[{"key":"2026042801401396664_j_forma-2021-0005_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_17.10.1007\/978-3-319-20615-8_17","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026042801401396664_j_forma-2021-0005_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\/s10817-017-9440-6.10.1007\/s10817-017-9440-6604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026042801401396664_j_forma-2021-0005_ref_003","unstructured":"[3] Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003."},{"key":"2026042801401396664_j_forma-2021-0005_ref_004","unstructured":"[4] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269\u2013275, 2004."},{"key":"2026042801401396664_j_forma-2021-0005_ref_005","unstructured":"[5] Laurent Schwartz. Th\u00e9orie des ensembles et topologie, tome 1. Analyse. Hermann, 1997."},{"key":"2026042801401396664_j_forma-2021-0005_ref_006","unstructured":"[6] Yasumasa Suzuki. Banach space of bounded real sequences. Formalized Mathematics, 12 (2):77\u201383, 2004."},{"key":"2026042801401396664_j_forma-2021-0005_ref_007","doi-asserted-by":"crossref","unstructured":"[7] Hiroshi Yamazaki. Functional sequence in norm space. Formalized Mathematics, 28(4): 263\u2013268, 2020. doi:10.2478\/forma-2020-0023.10.2478\/forma-2020-0023","DOI":"10.2478\/forma-2020-0023"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2021-0005","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T13:38:44Z","timestamp":1777383524000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2021-0005"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,1]]},"references-count":7,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,8,26]]},"published-print":{"date-parts":[[2021,4,1]]}},"alternative-id":["10.2478\/forma-2021-0005"],"URL":"https:\/\/doi.org\/10.2478\/forma-2021-0005","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,1]]}}}