Construction Of An n-Norm On \ell^p (R)

Mochammad Idris

doi:10.22342/jims.v32i2.2053

Mochammad Idris ⁽¹⁾

(1) Department of Mathematics, Universitas Lambung Mangkurat, Indonesia

https://doi.org/10.22342/jims.v32i2.2053

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
May 8, 2025

Accepted
November 22, 2025

Published
April 23, 2026

Keywords:

norm, n-norm, bounded linear functionals, $\ell^p$-spaces

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Abstract

In this article, we discuss an $n$-norm, with $n \ge 2$, which is defined through bounded linear functionals on $p$-summable sequence spaces. We also introduce a new norm induced by this $ n $-norm, which will be examined for its equivalence to the usual norm. Next, we demonstrate the relationships between various mappings, including the $ n $-norm, $ (\!n\!\!-\!\!1\!) $-norm, $\cdots,$ $ 2 $-norm, and the usual norm. Finally, our results show that the $ p $-summable sequence spaces, equipped with these mappings are complete spaces.

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References

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Authors

Mochammad Idris

Department of Mathematics, Universitas Lambung Mangkurat

moch.idris@ulm.ac.id (Primary Contact)

Idris, M. (2026). Construction Of An n-Norm On \ell^p (R). Journal of the Indonesian Mathematical Society, 32(2), 2053. https://doi.org/10.22342/jims.v32i2.2053

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1. INTRODUCTION

Let $X$ be a real vector space and $\| \cdot \|$ be a real function on $X$ satisfying the following three conditions:

$\| x\| \geq 0$ for every $x\in X$; $\| x\| = 0$ if and only if $x = 0\in X$;
$\|\alpha x\| = |\alpha|\|x\|$ for every $x \in X$ and for every $\alpha \in \mathbb{R}$;
$\lVert x+y\rVert \leq \lVert x\rVert+\lVert y\rVert$ for every $x,y \in X$.

The function $\|\cdot\|$ is called a norm on $X$ and the pair $(X,\|\cdot\|)$ is called a normed space[1]. A norm can be viewed as a generalization of the concept of length within a vector space.

A set ofp-summable sequences, with $1 \leq p \leq \infty$, is one of vector spaces in functional analysis. Here, the vector space is denoted by $\ell^{p} = \ell^{p}(\mathbb{R})$ which contains all sequences of real numbers $x = (x_{j})$ for which $\sum_{j=1}^{\infty}|x_{j}|^{p} < \infty$ (or $\sup_{j \in \mathbb{N}} |x_j| < \infty$ while $p = \infty$). Usually, $\ell^{p}$ is equipped with the usual norm

so that $\ell^p = (\ell^p, \| x \|_{\ell^p})$ becomes a normed space.

We also introduce ann-norm. Consider a vector space $X$ with $dim(X) \geq n$. By[2], we give a mapping $\|\cdot,\ldots,\cdot\|_{nX}: X \times \cdots \times X \longrightarrow \mathbb{R}$ such that it has four conditions:

$\|v_{1},\cdots,v_{n}\|_{nX}\geq0$ holds for every $v_{1},\ldots,v_{n}\in X$; and $v_{1},\ldots,v_{n}$ are linearly dependent if and only if $\|v_{1},\cdots,v_{n}\|_{nX}=0$;
$\|v_{1}, v_{2}, \cdots, v_{n}\|_{nX} = \|v_{i_{1}}, v_{i_{2}} \cdots, v_{i_{n}}\|_{nX}$ holds for every $v_{1}, v_{2} \ldots, v_{n} \in X$ and $(i_{1}, i_{2} \cdots, i_{n})$ is permutation of $(1, 2 \cdots, n)$;
$\|\alpha v_{1},\cdots,v_{n}\|_{nX}=|\alpha|\|\nu_{1},\cdots,\nu_{n}\|_{nX}$ holds for every $v_{1},\ldots,v_{n}\in X$ and for every scalar $\alpha\in R$;
$\|v_{1},\cdots,v_{n-1},y+z\|_{nX}\leq\|v_{1},\cdots,v_{n-1},y\|_{nX}+\|v_{1},\cdots,v_{n-1},z\|_{nX}$ holds for every $v_{1},\ldots,v_{n-1},y,z\in X$.

We call that $\|\cdot,\cdots,\cdot\|_{nX}$ is an n-norm and the pair $(X,\|\cdot,\cdots,\cdot\|_{nX})$ is ann-normed space. On $X$, for simplification, we occasionally express then-norm as

for every $v_{1},\cdots,v_{n}\in X$.

In functional analysis, the concept ofn-normed spaces generalizes normed spaces. This idea was studied by various researchers who examined different properties ofn-normed spaces. The notion ofn-normed spaces was first introduced by Misiak[3], and subsequent contributions were made by Dutta[4], Gunawan and Mashadi[5], Gunawan[6], Kim and Cho[7], Lewandowska[8], and many others. In the specific case wheren= 2, the notion of 2-normed spaces was introduced by Gähler in the mid-1960s (see[9]). This concept was further developed by Diminnie, Gähler, White [10,11], among others.

We can add an extra property to then-norm, as described in the following proposition.

Proposition 1.1.On$X$, we have a property

for every $v_{1},\ldots,v_{n}\in X$ and for every scalar $\alpha_{1},\cdots,\alpha_{n}\in \mathbb{R}$.

The proof of the above proposition is quite straightforward, relying only on the triangle inequality, the homogeneity property, and the concept of linearly dependent vectors. We leave the verification to the reader.

For $X = \ell^{p}$ where $1 \leq p < \infty$, we recalln-norm which is defined by Gunawan[2]. That is a mapping $\|\cdot, \cdots, \cdot\|_{n\ell^{p}} : \ell^{p} \times \cdots \times \ell^{p} \to \mathbb{R}$ which is stated by the following equation

for every $x_{1}, x_{2}, \cdots, x_{n} \in \ell^{p}$. For $p = \infty$, that is

for every $x_{1}, x_{2}, \cdots, x_{n} \in \ell^{\infty}$. Note that $\begin{Vmatrix}x_{1k_1} & x_{2k_1} & \cdots & x_{nk_1} \\x_{1k_2} & x_{2k_2} & \cdots & x_{nk_2} \\\vdots & \vdots & \ddots & \vdots \\x_{1k_n} & x_{2k_n} & \cdots & x_{nk_n}\end{Vmatrix}$ means an

absolute value of $det\left(\begin{array}{cccc}x_{1k_{1}} & x_{2k_{1}} & \cdots & x_{nk_{1}}\\ x_{1k_{2}} & x_{2k_{2}} & \cdots & x_{nk_{2}}\\ \vdots & \vdots & \ddots & \vdots \\ x_{1k_{n}} & x_{2k_{n}} & \cdots & x_{nk_{n}}\end{array}\right)$. In this article, then-norm defined by Gunawan (4) is referred to as the usualn-norm.

We now describe that a space $X$, equipped with a norm, is called a complete space (or Banach space) if every Cauchy sequence in $X$ converges to a limit within the space. In the context of ann-normed space $X$, a sequence $(x(m))$ converges to $x$ if and only if $\| v _ {1}, \ldots , v _ {(n - 1)}, x (m) - x \| _ {n X} \to 0$

for all $v_{1}, \ldots, v_{(n-1)} \in X$ as $m \to \infty$. Likewise, the sequence $(x(m))$ is Cauchy if and only if

for all $v_{1},\ldots,v_{(n-1)}\in X$ as $m_{1},m_{2}\to\infty$. Ann-Banach space is ann-normed space where every Cauchy sequence converges. It was shown in[1]and[2]that both the usual norm and the usualn-norm result in complete spaces, meaning that both are a Banach spaces and ann-Banach spaces respectively. On $\ell^{p}(\mathbb{R})$, Koncaet al.[12]conducted deeply investigation into the equivalence of two definitions of convergence (two definitions of Cauchy sequences).

In 2000, Gunawan introduced the norm induced by the usualn-norm and the firstnstandard basis vectors in $\ell^{p}(\mathbb{R})$[2]. Later, in 2013, he and his students refined this definition by using any set ofnlinearly independent vectors in $\ell^{p}(\mathbb{R})$[13]. This prompts the question: can we define ann-norm that is induced by a norm? For $p = 2$, defining a 2-norm on $\ell^{2}(\mathbb{R})$ is fairly straightforward, along with the corresponding 2-inner product. For $\ell^p (\mathbb{R})$ with $p\neq 2$, Konca et al. introduced a 2-inner product and

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