{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T01:27:12Z","timestamp":1783906032342,"version":"3.55.0"},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T00:00:00Z","timestamp":1648771200000},"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":[[2022,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In this article, Feed-forward Neural Network is formalized in the Mizar system [1], [2]. First, the multilayer perceptron [6], [7], [8] is formalized using functional sequences. Next, we show that a set of functions generated by these neural networks satisfies equicontinuousness and equiboundedness property [10], [5]. At last, we formalized the compactness of the function set of these neural networks by using the Ascoli-Arzela\u2019s theorem according to [4] and [3].<\/jats:p>","DOI":"10.2478\/forma-2022-0002","type":"journal-article","created":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T05:15:18Z","timestamp":1671599718000},"page":"13-21","source":"Crossref","is-referenced-by-count":1,"title":["Compactness of Neural Networks"],"prefix":"10.2478","volume":"30","author":[{"given":"Keiichi","family":"Miyajima","sequence":"first","affiliation":[{"name":"Ibaraki University , Faculty of Engineering , Hitachi, Ibaraki , Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hiroshi","family":"Yamazaki","sequence":"additional","affiliation":[{"name":"Nagano Prefectural Institute of Technology , Nagano , Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2022,12,21]]},"reference":[{"key":"2026071212490953053_j_forma-2022-0002_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":"2026071212490953053_j_forma-2022-0002_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":"2026071212490953053_j_forma-2022-0002_ref_003","doi-asserted-by":"crossref","unstructured":"[3] 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":"2026071212490953053_j_forma-2022-0002_ref_004","unstructured":"[4] Kazuo Matsuzaka. Sets and Topology (Introduction to Mathematics). IwanamiShoten, 2000."},{"key":"2026071212490953053_j_forma-2022-0002_ref_005","unstructured":"[5] Michael Read and Barry Simon. Functional Analysis (Methods of Modern Mathematical Physics). Academic Press, 1980."},{"key":"2026071212490953053_j_forma-2022-0002_ref_006","doi-asserted-by":"crossref","unstructured":"[6] Frank Rosenblatt. The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain. Psychological Review, 1958.10.1037\/h004251913602029","DOI":"10.1037\/h0042519"},{"key":"2026071212490953053_j_forma-2022-0002_ref_007","doi-asserted-by":"crossref","unstructured":"[7] David Everett Rumelhart, Geoffrey Everes Hinton, and Ronald J. Williams. Learning representations by backpropagating errors. Nature, 1986.10.1038\/323533a0","DOI":"10.1038\/323533a0"},{"key":"2026071212490953053_j_forma-2022-0002_ref_008","doi-asserted-by":"crossref","unstructured":"[8] J\u00fcrgen Schmidhuber. Deep Learning in Neural Networks: An Overview. Neural Networks, 2015.10.1016\/j.neunet.2014.09.00325462637","DOI":"10.1016\/j.neunet.2014.09.003"},{"key":"2026071212490953053_j_forma-2022-0002_ref_009","doi-asserted-by":"crossref","unstructured":"[9] Hiroshi Yamazaki, Keiichi Miyajima, and Yasunari Shidama. Ascoli-Arzel\u00e0 theorem. Formalized Mathematics, 29(2):87\u201394, 2021. doi:10.2478\/forma-2021-0009.","DOI":"10.2478\/forma-2021-0009"},{"key":"2026071212490953053_j_forma-2022-0002_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-2022-0002","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T01:03:49Z","timestamp":1783904629000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2022-0002"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,1]]},"references-count":10,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12,21]]},"published-print":{"date-parts":[[2022,4,1]]}},"alternative-id":["10.2478\/forma-2022-0002"],"URL":"https:\/\/doi.org\/10.2478\/forma-2022-0002","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,1]]}}}