Classification of the norming sets of $L_s(^3 l_1^2)$

Sung Guen Kim

Commentationes Mathematicae, Tome 61 (2023) no. 1-2, pp. 33-43.

Résumé

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 $\|x_1\|=\cdots=\|x_n\|=1$ and $|T(x_1, \ldots, x_n)|=\|T\|,$ where ${\mathcal L}(^n E)$ denotes the space of all continuous symmetric $n$-linear forms on $E.$ For $T\in {\mathcal L}(^n E),$ we define $$\qopname\relax o{Norm}(T)=\Big\{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n)~\mbox{is a norming point of}~T\Big\}.$$ $\qopname\relax o{Norm}(T)$ is called the {\em norming set} of $T$. In this paper, we classify $\qopname\relax o{Norm}(T)$ for every $T\in {\mathcal L}_s(^3 l_{1}^2),$ where ${\mathcal L}_s(^3 l_1^2)$ denotes the space of all continuous symmetric 3-linear forms on the plane with the $l_1$-norm.

Détail

DOI : 10.14708/cm.v61i1-2.7176