The norming sets of multilinear forms on a certain normed space $\mathbb{R}^n$

Abstract

 Let $n, m\in \mathbb{N}, n, m\geq 2$ and $E$ a Banach space. An element $(x_1, \ldots, x_n)\in E^n$ is called a~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  ${Norm}(T)$ as the set of all $(x_1, \ldots, x_n)\in E^n$ which are the norming points of~$T.$     Let $\mathbb{R}^n_{\|\cdot\|}=\mathbb{R}^n$ with a norm satisfying that $\{W_1, \ldots, W_n\}$ forms a basis and the set of all extreme points of $B_{\mathbb{R}^n_{\|\cdot\|}}$ is $\{\pm W_1, \ldots, \pm W_n\}$.   In the paper we characterize ${Norm}(T)$ for every $T\in {\mathcal L}(^m \mathbb{R}^n_{\|\cdot\|})$ as follows: Let $T=(T(W_{i_1}, \ldots W_{i_m}))_{\overset{1\leq i_k\leq n,}{1\leq k\leq m}}\in {\mathcal L}(^m \mathbb{R}^n_{\|\cdot\|})$, $\|T\|=1,$\ $S_T=(b_{i_1\cdots i_m})_{\overset{1\leq i_k\leq n,}{1\leq k\leq m}}\in {\mathcal L}(^m \mathbb{R}^n_{\|\cdot\|})$ such that   $\displaystyle b_{i_1\cdots i_m}=T\big(W_{i_1}, \ldots W_{i_m}\big)~\mbox{if}~ |T\big(W_{i_1}, \ldots W_{i_m}\big)|=1~ \mbox{and}~ b_{i_1\cdots i_m}=1~\mbox{if}~ |T\big(W_{i_1}, \ldots W_{i_m}\big)|<1,$   and $A$ is the Cartesian product of the set $\{1, \ldots, n\}$, $M$ is the set of indices $(i_1, \ldots, i_m)\in A$ such that $|T\big(W_{i_1}, \ldots W_{i_m}\big)|<1.$ Then, $$\begin{gather*} {Norm}(T)=\bigcap_{(i_1, \ldots, i_m)\in M} \bigcup_{j=1}^m \Big\{\Big( \sum_{1\leq i\leq n}s_i^{(1)}W_i, \ldots, \sum_{1\leq i\leq n}s_i^{(j-1)}W_i,  \sum_{1\leq i\leq n}s_i^{(j)}W_i-s_{i_j}W_{i_j}, \\ \sum_{1\leq i\leq n}s_i^{(j+1)}W_i, \ldots, \sum_{1\leq i\leq n}s_i^{(m)}W_i\Big)\colon    \Big(\sum_{1\leq i\leq n}s_i^{(1)}W_i, \ldots, \sum_{1\leq i\leq n}s_i^{(m)}W_i\Big)\in {Norm}(S_T)\Big\}. \end{gather*}$$