The norming sets of ${\mathcal L}\big({}^ml_{1}^n\big)$

Sung Guen Kim

doi:10.3842/umzh.v76i3.7294

Sung Guen Kim Department of Mathematics, Kyungpook National University, Republic of Korea

DOI: https://doi.org/10.3842/umzh.v76i3.7294

Keywords: norming sets, multilinear forms

Abstract

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 $m = 2.$

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