Norm attaining bilinear forms on the plane with the octagonal norm

Sung Guen Kim

Commentationes Mathematicae, Tome 61 (2023) no. 1-2, pp. 15-24.

Résumé

For given unit vectors $x_1, \cdots, x_n$ of a real Banach space $E,$ we define $$NA({\mathcal L}(^nE))(x_1, \cdots, x_n)=\{T\in {\mathcal L}(^nE): |T(x_1, \cdots, x_n)|=\|T\|=1\},$$ where ${\mathcal L}(^nE)$ denotes the Banach space of all continuous $n$-linear forms on $E$ endowed with the norm $\|T\|=\sup_{\|x_k\|=1, 1\leq k\leq n}{|T(x_1, \ldots, x_n)|}.$ In this paper, we classify $NA({\mathcal L}(^2 \mathbb{R}^2_{o(w)}))((x_1, x_2), (y_1, y_2))$ for unit vectors $(x_1, x_2), (y_1, y_2)\in \mathbb{R}^2_{o(w)},$ where $\mathbb{R}^2_{o(w)}=\mathbb{R}^2$ with the octagonal norm with weight $0

Détail

DOI : 10.14708/cm.v61i1-2.7128