{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T15:47:47Z","timestamp":1777477667320,"version":"3.51.4"},"reference-count":12,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2021,7,1]],"date-time":"2021-07-01T00:00:00Z","timestamp":1625097600000},"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":[[2021,7,1]]},"abstract":"<jats:p>\n                    <jats:bold>Summary<\/jats:bold>\n                    . In this article we formalize the Ascoli-Arzel\u00e0 theorem [5], [6], [8] in Mizar [1], [2]. First, we gave definitions of equicontinuousness and equiboundedness of a set of continuous functions [12], [7], [3], [9]. Next, we formalized the Ascoli-Arzel\u00e0 theorem using those definitions, and proved this theorem.\n                  <\/jats:p>","DOI":"10.2478\/forma-2021-0009","type":"journal-article","created":{"date-parts":[[2022,1,6]],"date-time":"2022-01-06T01:55:55Z","timestamp":1641434155000},"page":"87-94","source":"Crossref","is-referenced-by-count":2,"title":["Ascoli-Arzel\u00e0 Theorem"],"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 Faculty of Engineering Hitachi , 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":[[2021,12,30]]},"reference":[{"key":"2026042801422721756_j_forma-2021-0009_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":"2026042801422721756_j_forma-2021-0009_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":"2026042801422721756_j_forma-2021-0009_ref_3","unstructured":"Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003."},{"key":"2026042801422721756_j_forma-2021-0009_ref_4","doi-asserted-by":"crossref","unstructured":"Noboru Endou, Yasunari Shidama, and Katsumasa Okamura. Baire\u2019s category theorem and some spaces generated from real normed space. Formalized Mathematics, 14(4): 213\u2013219, 2006. doi:10.2478\/v10037-006-0024-x.10.2478\/v10037-006-0024-x","DOI":"10.2478\/v10037-006-0024-x"},{"key":"2026042801422721756_j_forma-2021-0009_ref_5","doi-asserted-by":"crossref","unstructured":"Serge Lang. Real and Functional Analysis (Texts in Mathematics). Springer-Verlag, 1993.10.1007\/978-1-4612-0897-6","DOI":"10.1007\/978-1-4612-0897-6"},{"key":"2026042801422721756_j_forma-2021-0009_ref_6","unstructured":"Kazuo Matsuzaka. Sets and Topology (Introduction to Mathematics). IwanamiShoten, 2000."},{"key":"2026042801422721756_j_forma-2021-0009_ref_7","unstructured":"Tohru Ozawa. Ascoli-Arzel\u00e0 theorem. 2012."},{"key":"2026042801422721756_j_forma-2021-0009_ref_8","unstructured":"Michael Read and Barry Simon. Functional Analysis (Methods of Modern Mathematical Physics). Academic Press, 1980."},{"key":"2026042801422721756_j_forma-2021-0009_ref_9","unstructured":"Laurent Schwartz. Th\u00e9orie des ensembles et topologie, tome 1. Analyse. Hermann, 1997."},{"key":"2026042801422721756_j_forma-2021-0009_ref_10","unstructured":"Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377\u2013380, 2003."},{"key":"2026042801422721756_j_forma-2021-0009_ref_11","doi-asserted-by":"crossref","unstructured":"Hiroshi Yamazaki, Keiichi Miyajima, and Yasunari Shidama. Functional space consisted by continuous functions on topological space. Formalized Mathematics, 29(1):49\u201362, 2021. doi:10.2478\/forma-2021-0005.10.2478\/forma-2021-0005","DOI":"10.2478\/forma-2021-0005"},{"key":"2026042801422721756_j_forma-2021-0009_ref_12","unstructured":"K\u00f4saku Yosida. Functional Analysis. 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