# Remarks on the norming sets of ${\mathcal L}(^ml_{1}^n)$ and description of the norming sets of ${\mathcal L}(^3l_{1}^2)$

• Sung Guen Kim Department of Mathematics, Kyungpook National University Daegu, Republic of Korea
Keywords: norming points; multilinear forms on $\mathbb{R}^n$ with $l_1$-norm

### Abstract

Let $n\in \mathbb{N}, n\geq 2.$ An element $x=(x_1, \ldots, x_n)\in E^n$ is called a {\em norming point} of $T\in {\mathcal L}(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and
$|T(x)|=\|T\|,$ 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 the {\em norming set} of $T$

$\mathop{Norm}(T)=\Big\{(x_1, \ldots, x_n)\in E^n: (x_1, \ldots, x_n)~\mbox{is a norming point of}~T\Big\}.$

By $i=(i_1,i_2,\ldots,i_m)$ we denote the multi-index. In this paper we show the following:

\noi (a) Let $n, m\geq 2$ and let $l_1^n=\mathbb{R}^n$ with the $l_1$-norm. Let $T=\big(a_{i}\big)_{1\leq i_k\leq n}\in {\mathcal L}(^ml_{1}^n)$ with $\|T\|=1.$
Define $S=\big(b_{i}\big)_{1\leq i_k\leq n}\in {\mathcal L}(^n l_1^m)$ be such that $b_{i}=a_{i}$ if
$|a_{i}|=1$ and $b_{i}=1$ if
$|a_{i}|<1.$

Let $A=\{1, \ldots, n\}\times \cdots\times\{1, \ldots, n\}$ and $M=\{i\in A: |a_{i}|<1\}.$
Then,

$\mathop{Norm}(T)=\bigcup_{(i_1, \ldots, i_m)\in M}\Big\{\Big(\big(t_1^{(1)}, \ldots, t_{{i_1}-1}^{(1)}, 0, t_{{i_1}+1}^{(1)}, \ldots, t_{n}^{(1)}\big), \big(t_1^{(2)}, \ldots, t_{n}^{(2)}\big), \ldots, \big(t_1^{(m)}, \ldots, t_{n}^{(m)}\big)\Big),$

$\Big(\big(t_1^{(1)}, \ldots, t_{n}^{(1)}\big), \big(t_1^{(2)}, \ldots, t_{{i_2}-1}^{(2)}, 0, t_{{i_2}+1}^{(2)}, \ldots, t_{n}^{(2)}\big), \big(t_1^{(3)}, \ldots, t_{n}^{(3)}\big), \ldots, \big(t_1^{(m)}, \ldots, t_{n}^{(m)}\big)\Big),\ldots$

$\ldots, \Big(\big(t_1^{(1)}, \ldots, t_{n}^{(1)}\big), \ldots, \big(t_1^{(m-1)}, \ldots, t_{n}^{(m-1)}\big), \big(t_1^{(m)}, \ldots, t_{{i_m}-1}^{(m)}, 0, t_{{i_m}+1}^{(m)}, \ldots, t_{n}^{(m)}\big)\Big)\colon$

$\Big(\big(t_1^{(1)}, \ldots, t_{n}^{(1)}\big), \ldots, \big(t_1^{(m)}, \ldots, t_{n}^{(m)}\big)\Big)\in \mathop{Norm}(S)\Big\}.$

This statement extend the results of [9].

\noi (b) Using the result (a), we describe the norming sets of every $T\in {\mathcal L}(^3l_{1}^2).$

### Author Biography

Sung Guen Kim, Department of Mathematics, Kyungpook National University Daegu, Republic of Korea

Department of Mathematics, Kyungpook National University
Daegu, Republic of Korea

### References

R.M. Aron, C. Finet, E. Werner, Some remarks on norm-attaining n-linear forms, Function spaces

(Edwardsville, IL, 1994), 19–28, Lecture Notes in Pure and Appl. Math., V.172, Dekker, New York,

E. Bishop, R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc., 67 (1961),

–98.

Y.S. Choi, S.G. Kim, Norm or numerical radius attaining multilinear mappings and polynomials, J.

London Math. Soc., 54 (1996), №2, 135–147.

S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, London, 1999.

M.J. Sevilla, R. Pay´a, Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence

spaces, Studia Math., 127 (1998), 99–112.

S.G. Kim, The norming set of a bilinear form on $l_{infty}^2$, Comment. Math., 60 (2020), №1–2, 37–63.

S.G. Kim, The norming set of a polynomial in ${mathcal P}(^2 l_{infty}^2)$, Honam Math. J., 42 (2020), №3, 569–576.

S.G. Kim, The norming set of a symmetric bilinear form on the plane with the supremum norm, Mat.

Stud., 55 (2021), №2, 171–180.

S.G. Kim, The norming set of a symmetric 3-linear form on the plane with the $l_1$-norm, New Zealand J.

Math., 51 (2021), 95–108.

S.G. Kim, The norming sets of ${mathcal L}(^2 l_1^2)$ and ${mathcal L}_s(^2 l_1^3),$, to appear in Bull. Transilv. Univ. Brasov, Ser. III:

Math. Copmut. Sci., 2(64) (2022), №2.

S.G. Kim, The norming sets of ${mathcal L}(^2 mathbb{R}^2_{h(w)})$, to appear in Acta Sci. Math. (Szeged), 89 (2023), №1–2.

Published
2023-01-16
How to Cite
Kim, S. G. (2023). Remarks on the norming sets of ${\mathcal L}(^ml_{1}^n)$ and description of the norming sets of ${\mathcal L}(^3l_{1}^2)$. Matematychni Studii, 58(2), 201-211. https://doi.org/10.30970/ms.58.2.201-211
Issue
Section
Articles