{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T23:07:27Z","timestamp":1783897647294,"version":"3.55.0"},"reference-count":7,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"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":[[2019,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In the first chapter, the notion of multilinear operator on real linear spaces is discussed. The algebraic structure [2] of multilinear operators is introduced here. In the second chapter, the results of the first chapter are extended to the case of the normed spaces. This chapter shows that bounded multilinear operators on normed linear spaces constitute the algebraic structure. We referred to [3], [7], [5], [6] in this formalization.<\/jats:p>","DOI":"10.2478\/forma-2019-0004","type":"journal-article","created":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T05:34:02Z","timestamp":1558071242000},"page":"35-45","source":"Crossref","is-referenced-by-count":1,"title":["Multilinear Operator and Its Basic Properties"],"prefix":"10.2478","volume":"27","author":[{"given":"Kazuhisa","family":"Nakasho","sequence":"first","affiliation":[{"name":"Yamaguchi University , Yamaguchi , Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2019,5,16]]},"reference":[{"key":"2026071214315520385_j_forma-2019-0004_ref_001_w2aab3b7b3b1b6b1ab1ab1Aa","unstructured":"[1] Czes\u0142aw Byli\u0144ski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661\u2013668, 1990."},{"key":"2026071214315520385_j_forma-2019-0004_ref_002_w2aab3b7b3b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Adam Grabowski, Artur Korni\u0142owicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363\u2013371, 2016. doi:10.15439\/2016F520.10.15439\/2016F520","DOI":"10.15439\/2016F520"},{"key":"2026071214315520385_j_forma-2019-0004_ref_003_w2aab3b7b3b1b6b1ab1ab3Aa","unstructured":"[3] Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972."},{"key":"2026071214315520385_j_forma-2019-0004_ref_004_w2aab3b7b3b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Marco Riccardi. Pocklington\u2019s theorem and Bertrand\u2019s postulate. Formalized Mathematics, 14(2):47\u201352, 2006. doi:10.2478\/v10037-006-0007-y.10.2478\/v10037-006-0007-y","DOI":"10.2478\/v10037-006-0007-y"},{"key":"2026071214315520385_j_forma-2019-0004_ref_005_w2aab3b7b3b1b6b1ab1ab5Aa","unstructured":"[5] Laurent Schwartz. Th\u00e9orie des ensembles et topologie, tome 1. Analyse. Hermann, 1997."},{"key":"2026071214315520385_j_forma-2019-0004_ref_006_w2aab3b7b3b1b6b1ab1ab6Aa","unstructured":"[6] Laurent Schwartz. Calcul diff\u00e9rentiel, tome 2. Analyse. Hermann, 1997."},{"key":"2026071214315520385_j_forma-2019-0004_ref_007_w2aab3b7b3b1b6b1ab1ab7Aa","unstructured":"[7] Kosaku Yoshida. Functional Analysis. Springer, 1980."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/27\/1\/article-p35.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2019-0004","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T22:51:28Z","timestamp":1783896688000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2019-0004"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,1]]},"references-count":7,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,5,16]]},"published-print":{"date-parts":[[2019,4,1]]}},"alternative-id":["10.2478\/forma-2019-0004"],"URL":"https:\/\/doi.org\/10.2478\/forma-2019-0004","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2019,4,1]]}}}