The Norming Set of a Bilinear Form on R² with the Octagonal Norm

S. G. Kim ; C. Y. Lee ; U. Jeong

Journal of convex analysis, Tome 30 (2023) no. 1, pp. 111-13

Résumé

An element $(x_1, \ldots, x_n)\in E^n$ is called {\em norming point} of $T\in {\mathcal L}(^n E)$ if\\[1mm] \centerline{$\|x_1\|=\cdots=\|x_n\|=1$ \ and \ $|T(x_1, \ldots, x_n)|=\|T\|$,}\\[1mm] 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\\[1mm] \centerline{$\text{\rm Norm\,}(T) = \{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n) \ \mbox{is a norming point of}\ T\}.$}\\[1mm] Let $\mathbb{R}^2_{o(w)}$ denote $\mathbb{R}^2$ with the octagonal norm with weight $0 < w\neq 1$\\[2mm] \centerline{$\|(x, y)\|_{o(w)}=\max\big\{|x|+w|y|, |y|+w|x|\big\}.$}\\[2mm] We classify $\text{\rm Norm\,}(T)$ for every $T\in {\mathcal L}(^2 \mathbb{R}_{o(w)}^2)$ with weight $0 < w\neq 1$ in this paper.

Détail

Classification : 46A22
Mots-clés : Norming points, bilinear forms