The norming set of a symmetric $n$-linear form on the plane with a rotated supremum norm for $n=3, 4, 5$
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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 $n$-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 $T$. Let $0\leq \theta\leq \frac{\pi}{4}$ and $\ell^2_{{\infty}, \theta}=\mathbb{R}^2$ with the rotated supremum norm $$\|(x, y)\|_{({\infty}, \theta)}=\max\Big\{|x \cos \theta+y \sin \theta|,~ |x \sin \theta-y \cos \theta|\Big\}.$$ In this paper, we characterize the norming set of $T\in {\mathcal L}(^n \ell_{(\infty, \theta)}^2)$. Using this result, we completely describe the norming set of $T\in {\mathcal L}_s(^n \ell_{(\infty, \theta)}^2)$ for $n=3, 4, 5$, where ${\mathcal L}_s(^n \ell_{(\infty, \theta)}^2)$ denotes the space of all continuous symmetric $n$-linear forms on $\ell_{(\infty, \theta)}^2$. We generalizes the results from \cite{9} for $n=3$ and $\theta=\frac{\pi}{4}$.
Abstract
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 $n$-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 $T$. Let $0\leq \theta\leq \frac{\pi}{4}$ and $\ell^2_{{\infty}, \theta}=\mathbb{R}^2$ with the rotated supremum norm $$\|(x, y)\|_{({\infty}, \theta)}=\max\Big\{|x \cos \theta+y \sin \theta|,~ |x \sin \theta-y \cos \theta|\Big\}.$$ In this paper, we characterize the norming set of $T\in {\mathcal L}(^n \ell_{(\infty, \theta)}^2)$. Using this result, we completely describe the norming set of $T\in {\mathcal L}_s(^n \ell_{(\infty, \theta)}^2)$ for $n=3, 4, 5$, where ${\mathcal L}_s(^n \ell_{(\infty, \theta)}^2)$ denotes the space of all continuous symmetric $n$-linear forms on $\ell_{(\infty, \theta)}^2$. We generalizes the results from \cite{9} for $n=3$ and $\theta=\frac{\pi}{4}$.
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