The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$
UDC 517.9 Let $n\in \mathbb{N},$ $n\geq 2.$ An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if\/ $\|x_1\| = \ldots = \|x_n\| = 1$ and $|T(x_1, \ldots, x_n)| = \|T\|, $ where ${\mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms...
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| Date: | 2024 |
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| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7294 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512646960250880 |
|---|---|
| author | Kim, Sung Guen Kim, Sung Guen |
| author_facet | Kim, Sung Guen Kim, Sung Guen |
| author_sort | Kim, Sung Guen |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-06-19T00:35:15Z |
| description | UDC 517.9
Let $n\in \mathbb{N},$ $n\geq 2.$ An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if\/ $\|x_1\| = \ldots = \|x_n\| = 1$ 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 \begin{align*}{\rm Norm}(T) = \big\{(x_1, \ldots, x_n)\in E^n\colon (x_1, \ldots, x_n) \mbox{is a norming point of} T\big\}.\end{align*} The ${\rm Norm}(T)$ is called the {\em norming set} of $T.$ For $m\in \mathbb{N},$ $m\geq 2, $ we characterize ${\rm Norm}(T)$ for every $T\in {\mathcal L}\big({}^m l_1^n\big),$ where $l_1^n = \mathbb{R}^n$ with the $l_1$-norm. As applications, we classify ${\rm Norm}(T)$ for every $T\in {\mathcal L}\big({}^m l_{1}^n\big)$ with $n = 2, 3$ and $m = 2.$ |
| doi_str_mv | 10.3842/umzh.v76i3.7294 |
| first_indexed | 2026-03-24T03:32:06Z |
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| id | umjimathkievua-article-7294 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:06Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-72942024-06-19T00:35:15Z The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$ The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$ Kim, Sung Guen Kim, Sung Guen norming sets multilinear forms UDC 517.9 Let $n\in \mathbb{N},$ $n\geq 2.$ An element $(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if\/ $\|x_1\| = \ldots = \|x_n\| = 1$ 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 \begin{align*}{\rm Norm}(T) = \big\{(x_1, \ldots, x_n)\in E^n\colon (x_1, \ldots, x_n) \mbox{is a norming point of} T\big\}.\end{align*} The ${\rm Norm}(T)$ is called the {\em norming set} of $T.$ For $m\in \mathbb{N},$ $m\geq 2, $ we characterize ${\rm Norm}(T)$ for every $T\in {\mathcal L}\big({}^m l_1^n\big),$ where $l_1^n = \mathbb{R}^n$ with the $l_1$-norm. As applications, we classify ${\rm Norm}(T)$ for every $T\in {\mathcal L}\big({}^m l_{1}^n\big)$ with $n = 2, 3$ and $m = 2.$ УДК 517.9 Нормуючі множини в ${\mathcal L}\big({}^ml_{1}^n\big)$ Нехай $n\in \mathbb{N},$ $n\geq 2. $  Елемент $(x_1, \ldots, x_n)\in E^n$ називається {\em нормуючою точкою} $T\in {\mathcal L}(^n E),$ якщо\/ $\|x_1\| = \ldots = \|x_n\| = 1$ і $|T(x_1, x_1, \ldots, x_n)| = \|T\|, $ де ${\mathcal L}(^n E)$ --- простір усіх неперервних $n$-лінійних форм на $E. $ Для $T\in {\mathcal L}(^n E)$ визначаємо \begin{align*}{\rm Norm}(T) = \big\{(x_1, \ldots, x_n)\in E^n\colon (x_1, \ldots, x_n) \mbox{--- точка нормування в} T\big\}.\end{align*} Множина ${\rm Norm}(T)$ називається {\em нормуючою множиною в} $T.$  Для $m\in \mathbb{N},$ $m\geq 2,$ ми характеризуємо ${\rm Norm}(T)$ для кожного $T\in {\mathcal L}\big({}^m l_1^n\big),$ де $l_1^n = \mathbb{R}^n$ з нормою $l_1$.  Як застосування, ми класифікуємо ${\rm Norm}(T)$ для кожного $T\in {\mathcal L}\big({}^m l_{1}^n\big)$ при $n = 2, 3$ і $m = 2.$ Institute of Mathematics, NAS of Ukraine 2024-03-25 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7294 10.3842/umzh.v76i3.7294 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 3 (2024); 382 - 394 Український математичний журнал; Том 76 № 3 (2024); 382 - 394 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7294/9856 Copyright (c) 2024 Sung Guen Kim |
| spellingShingle | Kim, Sung Guen Kim, Sung Guen The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$ |
| title | The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$ |
| title_alt | The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$ |
| title_full | The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$ |
| title_fullStr | The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$ |
| title_full_unstemmed | The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$ |
| title_short | The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$ |
| title_sort | norming sets of ${\mathcal l}\big({}^ml_{1}^n\big)$ |
| topic_facet | norming sets multilinear forms |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7294 |
| work_keys_str_mv | AT kimsungguen thenormingsetsofmathcallbigml1nbig AT kimsungguen thenormingsetsofmathcallbigml1nbig AT kimsungguen normingsetsofmathcallbigml1nbig AT kimsungguen normingsetsofmathcallbigml1nbig |