{"title":"实赋范向量空间上多线性映射空间与嵌套组合空间的同构","authors":"Kazuhisa Nakasho, Yuichi Futa","doi":"10.2478/forma-2022-0006","DOIUrl":null,"url":null,"abstract":"Summary This paper formalizes in Mizar [1], [2], that the isometric isomorphisms between spaces formed by an (n + 1)-dimensional multilinear map and an n-fold composition of linear maps on real normed spaces. This result is used to describe the space of nth-order derivatives of the Frechet derivative as a multilinear space. In Section 1, we discuss the spaces of 1-dimensional multilinear maps and 0-fold compositions as a preparation, and in Section 2, we extend the discussion to the spaces of (n + 1)-dimensional multilinear map and an n-fold compositions. We referred to [4], [11], [8], [9] in this formalization.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Isomorphism between Spaces of Multilinear Maps and Nested Compositions over Real Normed Vector Spaces\",\"authors\":\"Kazuhisa Nakasho, Yuichi Futa\",\"doi\":\"10.2478/forma-2022-0006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Summary This paper formalizes in Mizar [1], [2], that the isometric isomorphisms between spaces formed by an (n + 1)-dimensional multilinear map and an n-fold composition of linear maps on real normed spaces. This result is used to describe the space of nth-order derivatives of the Frechet derivative as a multilinear space. In Section 1, we discuss the spaces of 1-dimensional multilinear maps and 0-fold compositions as a preparation, and in Section 2, we extend the discussion to the spaces of (n + 1)-dimensional multilinear map and an n-fold compositions. We referred to [4], [11], [8], [9] in this formalization.\",\"PeriodicalId\":42667,\"journal\":{\"name\":\"Formalized Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2022-04-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Formalized Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2478/forma-2022-0006\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Formalized Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/forma-2022-0006","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Isomorphism between Spaces of Multilinear Maps and Nested Compositions over Real Normed Vector Spaces
Summary This paper formalizes in Mizar [1], [2], that the isometric isomorphisms between spaces formed by an (n + 1)-dimensional multilinear map and an n-fold composition of linear maps on real normed spaces. This result is used to describe the space of nth-order derivatives of the Frechet derivative as a multilinear space. In Section 1, we discuss the spaces of 1-dimensional multilinear maps and 0-fold compositions as a preparation, and in Section 2, we extend the discussion to the spaces of (n + 1)-dimensional multilinear map and an n-fold compositions. We referred to [4], [11], [8], [9] in this formalization.
期刊介绍:
Formalized Mathematics is to be issued quarterly and publishes papers which are abstracts of Mizar articles contributed to the Mizar Mathematical Library (MML) - the basis of a knowledge management system for mathematics.