{"title":"平面上具有$l_1$-范数的对称3-线性形式的赋范集","authors":"Sung Guen Kim","doi":"10.53733/177","DOIUrl":null,"url":null,"abstract":"An element $(x_1, \\ldots, x_n)\\in E^n$ is called a {\\em norming point} of $T\\in {\\mathcal L}(^n E)$ if $\\|x_1\\|=\\cdots=\\|x_n\\|=1$ and$|T(x_1, \\ldots, x_n)|=\\|T\\|,$ where ${\\mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$For $T\\in {\\mathcal L}(^n E),$ we define $${Norm}(T)=\\Big\\{(x_1, \\ldots, x_n)\\in E^n: (x_1, \\ldots, x_n)~\\mbox{is a norming point of}~T\\Big\\}.$$${Norm}(T)$ is called the {\\em norming set} of $T$. We classify ${Norm}(T)$ for every $T\\in {\\mathcal L}_s(^3 l_{1}^2)$.\n ","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"12 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"The norming set of a symmetric 3-linear form on the plane with the $l_1$-norm\",\"authors\":\"Sung Guen Kim\",\"doi\":\"10.53733/177\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"An element $(x_1, \\\\ldots, x_n)\\\\in E^n$ is called a {\\\\em norming point} of $T\\\\in {\\\\mathcal L}(^n E)$ if $\\\\|x_1\\\\|=\\\\cdots=\\\\|x_n\\\\|=1$ and$|T(x_1, \\\\ldots, x_n)|=\\\\|T\\\\|,$ where ${\\\\mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$For $T\\\\in {\\\\mathcal L}(^n E),$ we define $${Norm}(T)=\\\\Big\\\\{(x_1, \\\\ldots, x_n)\\\\in E^n: (x_1, \\\\ldots, x_n)~\\\\mbox{is a norming point of}~T\\\\Big\\\\}.$$${Norm}(T)$ is called the {\\\\em norming set} of $T$. We classify ${Norm}(T)$ for every $T\\\\in {\\\\mathcal L}_s(^3 l_{1}^2)$.\\n \",\"PeriodicalId\":30137,\"journal\":{\"name\":\"New Zealand Journal of Mathematics\",\"volume\":\"12 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-12-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"New Zealand Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.53733/177\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"New Zealand Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.53733/177","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
The norming set of a symmetric 3-linear form on the plane with the $l_1$-norm
An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and$|T(x_1, \ldots, x_n)|=\|T\|,$ where ${\mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$For $T\in {\mathcal L}(^n E),$ we define $${Norm}(T)=\Big\{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n)~\mbox{is a norming point of}~T\Big\}.$$${Norm}(T)$ is called the {\em norming set} of $T$. We classify ${Norm}(T)$ for every $T\in {\mathcal L}_s(^3 l_{1}^2)$.