A New Semi-Inner Product and p_n-Angle in the Space of p-Summable Sequences

Abstract

In this paper, we propose a definition for a semi-inner product in the space of p-summable sequences equipped with an n-norm. Using this definition, we introduce the concepts of $p_n$-orthogonality and the $p_n$-angle between two vectors in the space of p-summable sequences. For the special case n = 1, these concepts are identical to previous studies. We also introduce the notion of the $p_n$-angle between one-dimensional subspaces and arbitrary-dimensional subspaces. The authors believe that the results obtained in this paper are very significant, especially in the theory of n-normed space in functional analysis.

1. Introduction

Let X be the vector space. A semi-inner product on X is a mapping $[·,·]:X^2\to \mathbb{R}$, which satisfies the following properties:

(1)

$[x,x]\ge 0$ for every $x\in X$ and $[x,x]=0$ if and only if $x=0$;

(2)

$[\alpha x,\beta y]=\alpha \beta ·[x,y]$ for every $x,y\in X$ and $\alpha ,\beta \in \mathbb{R}$;

(3)

$[x+y,z]=[x,z]+[y,z]$ for every $x,y,z\in X$;

(4)

$\left|[x,y]\right|\le {[x,x]}^{\frac{1}{2}}{[y,y]}^{\frac{1}{2}}$ for every $x,y\in X$.

The pair $(X,[·,·\left]\right)$ is called a semi-inner product space. In this space, we can define a norm, that is, $\parallel ·\parallel ={[·,·]}^{\frac{1}{2}}$ [1].

Let $(X,\parallel ·\parallel )$ be a normed space. As it is known, not all normed spaces are inner product spaces, but we can define the semi-inner product. For instance, the space ${\ell }^p$ for $1\le p<\infty$, which is the space of all the p-summable sequences with norm ${∥x∥}_p={\left[{\displaystyle \sum _{n=1}^{\infty }}{\left|x_n\right|}^p\right]}^{\frac{1}{p}}$, is not an inner product space, except for $p=2$. Konca et al. [2] define a (weighted) inner product ${⟨·,·⟩}_v$ and a weighted norm on ${\ell }^p$. In this space with usual norm, we may check that

$$[x,y]={∥y∥}_p^{2-p}\sum _j{\left|y_j\right|}^{p-1}\mathrm{sgn}\left(y_j\right)x_j,x:=\left(x_j\right),y:=\left(y_j\right)\in l^p$$

(1)

is a semi-inner product on $({\ell }^p,\parallel ·{\parallel }_p)$ for $1\le p<\infty$ [3].

Next, the mapping $g:X^2\to \mathbb{R}$ defined by the formula

$$g(x,y):=\frac{1}{2}∥x∥\left[{\tau }_{+}(x,y)+{\tau }_{-}(x,y)\right],$$

with

$${\tau }_{\pm }(x,y):=\underset{t\to \pm 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

is the semi-inner product on X if $g(x,y)$ is linear in y.

Using the concept of semi-inner product g, Miličić [4] introduced the notions of g-orthogonality, namely $x{\perp }_{y}y$ if and only if $g(x,y)=0$ and the g-angle, namely $A_g(x,y):=\arccos \frac{g(x,y)}{\parallel x\parallel \parallel y\parallel }$. Many researchers have studied the g-orthogonal and g-angle between two vectors and two subspaces in X; see, for example, [5,6,7,8]. In 2018, Nur et al. [9] developed the notion of the g-angle between two subspaces. If $V=\mathrm{span}\left\{v\right\}$ and $W=\mathrm{span}\{w_1,\cdots ,w_m\}$ of X with $m\ge 1$, then the g-angle between V and W is defined by $A_g(V,W)$ with ${\cos }^2{A}_g(V,W)=\frac{\parallel v_W{\parallel }^2}{{\parallel v\parallel }^2}$. Here, $v_W$ denotes the g-orthogonal projection of v on W. Recently, Nur et al. [10] defined the standard n-norm using the (weighted) inner product and discussed the angle between two subspaces in the space of the p-summable.

In general, an n-norm on a real vector space X is a mapping $\parallel ·,\dots ,·\parallel :X\times \cdots \times X⟶\mathbb{R}$, which satisfies the following four conditions:

(1)

$\parallel x_1,\dots ,x_n\parallel =0$ if and only if $x_1,\dots ,x_n$ are linearly dependent;

(2)

$\parallel x_1,\dots ,x_n\parallel$ is invariant under permutation;

(3)

$\parallel \alpha x_1,\dots ,x_n\parallel =\left|\alpha \right|\parallel x_1,\dots ,x_n\parallel$ for every $x_1,\dots ,x_n\in X$ and for every $\alpha \in \mathbb{R}$;

(4)

$\parallel x_1,\dots ,x_{n-1},y+z\parallel \le \parallel x_1,\dots ,x_{n-1},y\parallel +\parallel x_1,\dots ,x_{n-1},z\parallel$ for every $x,y,z\in X$.

The pair $(X,∥·,\dots ,·∥)$ is called an n-normed space.

Geometrically, $\parallel x_1,\dots ,x_n\parallel$ may be interpreted as the volume of the n-dimensional parallelepiped spanned by $x_1,\dots ,x_n$. The theory of n-normed spaces for $n\ge 2$ was developed in the late 1960s [11,12,13]. Recent results can be found, for example, in [14,15,16]. On the space ${\ell }^p$ for $1\le p<\infty$, the following n-norm was defined by Gunawan in [17]:

$$\parallel x_1,\dots ,x_n{\parallel }_p:={\left[\frac{1}{n!}\sum _{k_1}\cdots \sum _{k_n}{\left(\mathrm{abs}\left|\begin{array}{ccc}{x}_{1k_1}& \cdots & x_{1k_n}\\ ⋮& \ddots & ⋮\\ x_{n{k}_1}& \cdots & x_{n{k}_n}\end{array}\right|\right)}^p\right]}^{\frac{1}{p}}.$$

(2)

The aim of this paper is to define a semi-inner product in an n-normed space $({\ell }^p,\parallel ·,\dots ,·{\parallel }_p)$ with $1\le p<\infty$. Using this result, we can introduce the $p_n$-orthogonal and the $p_n$-angle between two vectors. We also will discuss their properties. Moreover, we will define the $p_n$-angle between a one-dimensional subspace and an m-dimensional subspace in the n-normed space $({\ell }^p,\parallel ·,\cdots ,·{\parallel }_p)$.

2. Main Results

2.1. Semi-Inner Product and $p_n$-Angle between Two Vectors

In this subsection, we shall begin with the new semi-inner product on ${\ell }^p$ spaces equipped with an n-norm. Let $({\ell }^p,\parallel ·,\dots ,·\parallel )$ be an n-normed space and $\{a_1,\dots ,a_n\}$ be a linearly independent set on ${\ell }^p$. Now, we define the following mapping.

$${\parallel x\parallel }_{p_n}:={\left[\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}{\parallel x,a_{i_2},\dots ,a_{i_n}\parallel }_p^p\right]}^{\frac{1}{p}},$$

(3)

for every $x\in {\ell }^p$. Next, we have the following proposition.

Proposition 1

([17]). The mapping ${\parallel ·\parallel }_{p_n}$ defines a norm on ${\ell }^p$.

Example 1.

For $p=1,n=3$. Let $\{a_1,a_2,a_3\}$ be the Schauder basis on ${\ell }^1$, that is, $a_i=\left({\delta }_{ij}\right),i=1,2,3$ and $x=(2,1,3,0,\cdots )$. We observe that

$$\begin{array}{cc}\hfill {\parallel x\parallel }_{1_3}=& \parallel x,a_1,a_2{\parallel }_1+\parallel x,a_1,a_3{\parallel }_1+{\parallel x,a_2,a_3\parallel }_1\hfill \\ \hfill =& \frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{x}_{k_1}& x_{k_2}& x_{k_3}\\ a_{1k_1}& a_{1k_2}& a_{1k_3}\\ a_{2k_1}& a_{2k_2}& a_{2k_3}\end{array}\right|\right)+\frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{x}_{k_1}& x_{k_2}& x_{k_3}\\ a_{1k_1}& a_{1k_2}& a_{1k_3}\\ a_{3k_1}& a_{3k_2}& a_{3k_3}\end{array}\right|\right)\hfill \\ \hfill & +\frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{2k_1}& a_{2k_2}& a_{2k_3}\\ a_{3k_1}& a_{3k_2}& a_{3k_3}\end{array}\right|\right)=3+1+2=6.\hfill \end{array}$$

Using the norm ${\parallel ·\parallel }_{p_n}$ with $a_i=\left(a_{ij}\right)$ for $i=1,\dots ,n$, we define a mapping ${[·,·]}_{p_n}$ on the n-normed space $(l^p,\parallel ·,\dots ,·{\parallel }_p)$ with $1\le p<\infty$ by

$$\begin{array}{cc}\hfill {[x,y]}_{p_n}& =\left[\frac{{\left({∥x∥}_{p_n}\right)}^{2-p}}{n!}\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}\sum _{k_1}\cdots \sum _{k_n}{\left(\mathrm{abs}\left|\begin{array}{ccc}{x}_{1k_1}& \cdots & x_{1k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\right)}^{p-1}\times \right.\hfill \\ \hfill & \times \left.\mathrm{sgn}\left|\begin{array}{ccc}{x}_{k_1}& \cdots & x_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\left|\begin{array}{ccc}{y}_{k_1}& \cdots & y_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\right].\hfill \end{array}$$

(4)

for every $x=\left(x_j\right),y=\left(y_j\right)\in {\ell }^p.$

Then we have the following result.

Theorem 1.

The mapping ${[x,y]}_{p_n}$ in (4) defines a semi-inner product on $({\ell }^p,\parallel ·,\dots ·{\parallel }_p)$.

Proof. 

We will verify that ${[x,y]}_{p_n}$ satisfies the properties (1–4) of the semi-inner product.

• Observe that

  $$\begin{array}{cc}\hfill {[x,x]}_{p_n}& ={\left({∥x∥}_{p_n}\right)}^{2-p}\left[\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}{\parallel x,a_{i_2},\dots ,a_{i_n}\parallel }_p^p\right]\hfill \\ \hfill & ={\left({∥x∥}_{p_n}\right)}^2.\hfill \end{array}$$
• Observe that

  $$\begin{array}{cc}\hfill {[\alpha x,\beta y]}_{p_n}=& \left[\frac{{\left({∥\alpha x∥}_{p_n}\right)}^{2-p}}{n!}\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}\sum _{k_1}\cdots \sum _{k_n}{\left(\mathrm{abs}\left|\begin{array}{ccc}\alpha x_{1k_1}& \cdots & \alpha x_{1k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\right)}^{p-1}\times \right.\hfill \\ \hfill & \times \left.\mathrm{sgn}\left|\begin{array}{ccc}\alpha x_{k_1}& \cdots & \alpha x_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\left|\begin{array}{ccc}\beta y_{k_1}& \cdots & \beta y_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\right]\hfill \\ \hfill =& \alpha \beta {[x,y]}_{p_n}.\hfill \end{array}$$
• Using the properties of the determinant, we have

  $$\begin{array}{cc}\hfill {[x,y+y^{\prime }]}_{p_n}=& \left[\frac{{\left({∥x∥}_{p_n}\right)}^{2-p}}{n!}\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}\sum _{k_1}\cdots \sum _{k_n}{\left(\mathrm{abs}\left|\begin{array}{ccc}{x}_{k_1}& \cdots & x_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\right)}^{p-1}\times \right.\hfill \\ \hfill & \times \left.\mathrm{sgn}\left|\begin{array}{ccc}{x}_{k_1}& \cdots & x_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\left|\begin{array}{ccc}{y}_{k_1}+y_{k_1}^{\prime }& \cdots & y_{k_n}+y_{k_n}^{\prime }\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\right]\hfill \\ \hfill =& {[x,y]}_{p_n}+{[x,y^{\prime }]}_{p_n}.\hfill \end{array}$$
• Observe that

  $$\begin{array}{cc}\hfill \left|{[x,y]}_{p_n}\right|& \le \left[\frac{{\left({∥x∥}_{p_n}\right)}^{2-p}}{n!}\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}\sum _{k_1}\cdots \sum _{k_n}{\left(\mathrm{abs}\left|\begin{array}{ccc}{x}_{k_1}& \cdots & x_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\right)}^{p-1}\times \right.\hfill \\ \hfill & \times \left.\mathrm{abs}\left|\begin{array}{ccc}{y}_{k_1}& \cdots & y_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\right].\hfill \\ \hfill & \le \left[{\left({∥x∥}_{p_n}\right)}^{2-p}{\left[\frac{1}{n!}\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}\sum _{k_1}\cdots \sum _{k_n}{\left(\mathrm{abs}\left|\begin{array}{ccc}{x}_{k_1}& \cdots & x_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\right)}^p\right]}^{\frac{p-1}{p}}\times \right.\hfill \\ \hfill & \times \left.{\left[\frac{1}{n!}\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}\sum _{k_1}\cdots \sum _{k_n}{\left(\mathrm{abs}\left|\begin{array}{ccc}{y}_{k_1}& \cdots & y_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_n}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_n}\end{array}\right|\right)}^p\right]}^{\frac{1}{p}}\right].\hfill \\ \hfill & ={∥x∥}_{p_n}{∥y∥}_{p_n}.\hfill \end{array}$$

Therefore, ${[x,y]}_{p_n}$ defines a semi-inner product on ${\ell }^p$. □

Example 2.

For $p=1,n=3$. Let $\{a_1,a_2,a_3\}$ be the Schauder basis on ${\ell }^1$, that is, $a_i=\left({\delta }_{ij}\right),i=1,2,3$ and ${\parallel ·\parallel }_{1_3}$ in Proposition 1. If $x=(2,1,3,0,\cdots )$ and $y=(1,2,-1,0,\cdots )$ in ${\ell }^1$, we have

$$\begin{array}{cc}\hfill {\parallel x\parallel }_{1_3}=& \parallel x,a_1,a_2{\parallel }_1+\parallel x,a_1,a_3{\parallel }_1+{\parallel x,a_2,a_3\parallel }_1\hfill \\ \hfill =& \frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{x}_{k_1}& x_{k_2}& x_{k_3}\\ a_{1k_1}& a_{1k_2}& a_{1k_3}\\ a_{2k_1}& a_{2k_2}& a_{2k_3}\end{array}\right|\right)+\frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{x}_{k_1}& x_{k_2}& x_{k_3}\\ a_{1k_1}& a_{1k_2}& a_{1k_3}\\ a_{3k_1}& a_{3k_2}& a_{3k_3}\end{array}\right|\right)\hfill \\ \hfill & +\frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{2k_1}& a_{2k_2}& a_{2k_3}\\ a_{3k_1}& a_{3k_2}& a_{3k_3}\end{array}\right|\right)=3+1+2=6.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\parallel y\parallel }_{1_3}=& \parallel y,a_1,a_2{\parallel }_1+\parallel y,a_1,a_3{\parallel }_1+{\parallel y,a_2,a_3\parallel }_1\hfill \\ \hfill =& \frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{1k_1}& a_{1k_2}& a_{1k_3}\\ a_{2k_1}& a_{2k_2}& a_{2k_3}\end{array}\right|\right)+\frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{1k_1}& a_{1k_2}& a_{1k_3}\\ a_{3k_1}& a_{3k_2}& a_{3k_3}\end{array}\right|\right)\hfill \\ \hfill & +\frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{2k_1}& a_{2k_2}& a_{2k_3}\\ a_{3k_1}& a_{3k_2}& a_{3k_3}\end{array}\right|\right)=1+2+1=4\hfill \end{array}$$

As a consequence,

$$\begin{array}{cc}\hfill {[x,y]}_{1_3}=& \frac{\left({∥x∥}_{1_3}\right)}{3!}\sum _{\{i_2,i_3\}\subset \{1,2,3\}}\sum _{k_1}\sum _{k_2}\sum _{k_3}sgn\left|\begin{array}{ccc}{x}_{k_1}& x_{k_2}& x_{k_3}\\ a_{i_2{k}_1}& a_{i_2{k}_2}& a_{i_2{k}_3}\\ a_{i_3{k}_1}& a_{i_3{k}_2}& a_{i_3{k}_3}\end{array}\right|\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{i_2{k}_1}& a_{i_2{k}_2}& a_{i_2{k}_3}\\ a_{i_3{k}_1}& a_{i_3{k}_2}& a_{i_3{k}_3}\end{array}\right|\hfill \\ \hfill =& -6+12+6=12\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {[y,x]}_{1_3}=& \frac{\left({∥y∥}_{1_3}\right)}{3!}\sum _{\{i_2,i_3\}\subset \{1,2,3\}}\sum _{k_1}\sum _{k_2}\sum _{k_3}sgn\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{i_2{k}_1}& a_{i_2{k}_2}& a_{i_2{k}_3}\\ a_{i_3{k}_1}& a_{i_3{k}_2}& a_{i_3{k}_3}\end{array}\right|\left|\begin{array}{ccc}{x}_{k_1}& x_{k_2}& x_{k_3}\\ a_{i_2{k}_1}& a_{i_2{k}_2}& a_{i_2{k}_3}\\ a_{i_3{k}_1}& a_{i_3{k}_2}& a_{i_3{k}_3}\end{array}\right|\hfill \\ \hfill =& -18+6+12=0.\hfill \end{array}$$

Remark 1.

The function ${[x,y]}_{1_3}$ does not satisfy commutative property. In the example above, we observe that ${[x,y]}_{1_3}=12\ne {[y,x]}_{1_3}=0$.

Specifically for $p=2$, we observe that

$$\begin{array}{cc}\hfill {[x,y]}_{2_n}& =\frac{1}{n!}\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}\sum _{k_1}\cdots \sum _{k_n}\left|\begin{array}{ccc}{x}_{k_1}& \cdots & x_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_1}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_1}\end{array}\right|\left|\begin{array}{ccc}{y}_{k_1}& \cdots & y_{k_n}\\ a_{i_2{k}_1}& \cdots & a_{i_2{k}_1}\\ ⋮& \ddots & ⋮\\ a_{i_n{k}_1}& \cdots & a_{i_n{k}_1}\end{array}\right|\hfill \\ \hfill & ={[y,x]}_{2_n}.\hfill \end{array}$$

As consequence, we have following corollary:

Corollary 1.

The mapping ${[x,y]}_{2_n}$ in (4) defines an inner product on $({\ell }^2,\parallel ·,\dots ·{\parallel }_2)$.

Example 3.

For $n=2$. Let $\{a_1,a_2\}$ be the Schauder basis on ${\ell }^2$, that is, $a_i=\left({\delta }_{ij}\right),i=1,2$. If $x=(2,1,0,0,\cdots )$ and $y=(1,2,0,0,\cdots )$ in ${\ell }^2$, we have

$$\begin{array}{cc}\hfill {[x,y]}_{2_2}=& \frac{1}{2}\sum _{\left\{i_2\right\}\subset \{1,2\}}\sum _{k_1}\sum _{k_2}\left|\begin{array}{cc}{x}_{k_1}& x_{k_2}\\ a_{i_2{k}_1}& a_{i_2{k}_2}\end{array}\right|\left|\begin{array}{cc}{y}_{k_1}& y_{k_2}\\ a_{i_2{k}_1}& a_{i_2{k}_2}\end{array}\right|\hfill \\ \hfill =& \frac{1}{2}\sum _{k_1}\sum _{k_2}\left|\begin{array}{cc}{x}_{k_1}& x_{k_2}\\ a_{1k_1}& a_{1k_2}\end{array}\right|\left|\begin{array}{cc}{y}_{k_1}& y_{k_2}\\ a_{1k_1}& a_{1k_2}\end{array}\right|+\frac{1}{2}\sum _{k_1}\sum _{k_2}\left|\begin{array}{cc}{x}_{k_1}& x_{k_2}\\ a_{2k_1}& a_{2k_2}\end{array}\right|\left|\begin{array}{cc}{y}_{k_1}& y_{k_2}\\ a_{2k_1}& a_{2k_2}\end{array}\right|\hfill \\ \hfill =& 4.\hfill \end{array}$$

By using the semi-inner product ${[x,y]}_{p_n}$, we define orthogonality in $({\ell }^p,\parallel ·,\dots ·{\parallel }_n)$ as follows:

Definition 1

($p_n$-orthogonality). Let $({\ell }^p,\parallel ·,\dots ·{\parallel }_n)$ be the n-normed space. Vector x is $p_n$-orthogonal to vector y, and we write $x{\perp }_{p_n}y$ if and only if ${[x,y]}_{p_n}=0$.

Example 4.

For $p=1,n=3$. Let $\{a_1,a_2,a_3\}$ be the Schauder basis on ${\ell }^1$, that is, $a_i=\left({\delta }_{ij}\right),i=1,2,3$. If $x=(1,2,-1,0,\cdots )$ and $y=(2,1,3,0,\cdots )$ in ${\ell }^1$, using Example 2, we have ${[x,y]}_{p_n}=0$. Hence, x is $1_3$-orthogonal to y.

Next, $p_n$-orthogonality has the following properties.

Proposition 2.

The $p_n$-orthogonality satisfies the following properties:

(a) 

Nondegeneracy property: If $x{\perp }_{p_n}x$, then $x=0$.

(b) 

Homogeneity property: If $x{\perp }_{p_n}y$, then $\alpha x{\perp }_{p_n}\beta y$ for every $\alpha ,\beta \in \mathbb{R}$.

(c) 

Right additive property: If $x{\perp }_{p_n}y$ and $x{\perp }_{p_n}z$, then $x{\perp }_g(y+z)$.

(d) 

Resolvability property: For every $x,y\in X$ there is $\gamma \in \mathbb{R}$ such that $x{\perp }_{p_n}(\gamma x+y)$.

Proof. 

By using Theorem 1, the properties (a)−(c) are obviously true.

(d)

Let $x,y\in {\ell }^p$. For case $x=0$, the resolvability property is fulfilled. Next, for case $x\ne 0$, choose $\gamma =-\frac{{[x,y]}_{p_n}}{{(\parallel x\parallel }_{p_n}{)}^2}$. Using Theorem 1, we have

$$\begin{array}{cc}\hfill {[x,\gamma x+y]}_{p_n}& =\frac{1}{\gamma }{[\gamma x,\gamma x+y]}_{p_n}\hfill \\ \hfill & =\frac{1}{\gamma }{(\gamma \parallel x\parallel }_{p_n}^2+{[\gamma x,y]}_{p_n})\hfill \\ \hfill & ={\gamma \parallel x\parallel }_{p_n}^2+{[x,y]}_{p_n}=0,\hfill \end{array}$$

as desired.

 □

Remark 2.

Note that for $n=1$, the $p_1$-orthogonality coincides with the g-orthogonality in ${\ell }^p$. Specifically for $p=2$, the $2_n$-orthogonality satisfies the symmetry and continuity property. Next, by using Remark 1, the $p_n$-orthogonality does not satisfy the symmetry property. The $p_n$-orthogonality also does not satisfy continuity property.

Example 5.

For $p=1$. Take $x_k=(\frac{1}{k},1,0,\cdots )$, $x=(0,1,0,\cdots )$, and $y=(1,1,0,..)$ in ${\ell }^1$. Using inequality ${\parallel x\parallel }_1\le {\parallel x\parallel }_{1_n}\le n{\parallel x\parallel }_1$ in [17], we have $x_k\to x$ (in norm ${\parallel ·\parallel }_{1_n}$). Next, we observe that ${[x_k,y]}_{1_n}=0$ for every $k\in \mathbb{N}$, but ${[x,y]}_{1_n}\ne 0$.

Using a semi-inner product in (4), we define the angle between two nonzero vectors x and y on $({\ell }^p,\parallel ·,\dots ,·{\parallel }_p)$ as follows:

$$A_{p_n}(x,y):=\arccos \frac{{[y,x]}_{p_n}}{{\parallel x\parallel }_{p_n}{\parallel y\parallel }_{p_n}}.$$

(5)

Note that $y{\perp }_{p_n}x$ if and only if $A_{p_n}(x,y)=\frac{1}{2}\pi .$ We can observe that the angle $A_{p_n}(x,y)$ for $n=1$ is identical with the g-angle $A_g(x,y)$ in [9].

Example 6.

Let $({\ell }^1,\parallel ·,·,·{\parallel }_1)$ be 3 normed space and $\{a_1,a_2,a_3\}$ be the Schauder basis on ${\ell }^1$, that is, $a_i=\left({\delta }_{ij}\right),i=1,2,3$ and ${\parallel ·\parallel }_{1_3}$ in Proposition 1. If $x=(1,2,-1,0,\cdots )$ and $y=(2,1,3,0,\cdots )$ in ${\ell }^1$, we observe that

$$\begin{array}{cc}\hfill {\parallel x\parallel }_{1_3}=& \parallel x,a_1,a_2{\parallel }_1+\parallel x,a_1,a_3{\parallel }_1+{\parallel x,a_2,a_3\parallel }_1\hfill \\ \hfill =& \frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{x}_{k_1}& x_{k_2}& x_{k_3}\\ a_{1k_1}& a_{1k_2}& a_{1k_3}\\ a_{2k_1}& a_{2k_2}& a_{2k_3}\end{array}\right|\right)+\frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{x}_{k_1}& x_{k_2}& x_{k_3}\\ a_{1k_1}& a_{1k_2}& a_{1k_3}\\ a_{3k_1}& a_{3k_2}& a_{3k_3}\end{array}\right|\right)\hfill \\ \hfill & +\frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{x}_{k_1}& x_{k_2}& x_{k_3}\\ a_{2k_1}& a_{2k_2}& a_{2k_3}\\ a_{3k_1}& a_{3k_2}& a_{3k_3}\end{array}\right|\right)=1+2+1=4\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\parallel y\parallel }_{1_3}=& \parallel y,a_1,a_2{\parallel }_1+\parallel y,a_1,a_3{\parallel }_1+{\parallel y,a_2,a_3\parallel }_1\hfill \\ \hfill =& \frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{1k_1}& a_{1k_2}& a_{1k_3}\\ a_{2k_1}& a_{2k_2}& a_{2k_3}\end{array}\right|\right)+\frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{1k_1}& a_{1k_2}& a_{1k_3}\\ a_{3k_1}& a_{3k_2}& a_{3k_3}\end{array}\right|\right)\hfill \\ \hfill & +\frac{1}{6}\sum _{k_1}\sum _{k_2}\sum _{k_3}\left(abs\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{2k_1}& a_{2k_2}& a_{2k_3}\\ a_{3k_1}& a_{3k_2}& a_{3k_3}\end{array}\right|\right)=3+1+2=6.\hfill \end{array}$$

As a consequence,

$$\begin{array}{cc}\hfill {[y,x]}_{1_3}=& \frac{\left({∥y∥}_{1_3}\right)}{3!}\sum _{\{i_2,i_3\}\subset \{1,2,3\}}\sum _{k_1}\sum _{k_2}\sum _{k_3}sgn\left|\begin{array}{ccc}{y}_{k_1}& y_{k_2}& y_{k_3}\\ a_{i_2{k}_1}& a_{i_2{k}_2}& a_{i_2{k}_3}\\ a_{i_3{k}_1}& a_{i_3{k}_2}& a_{i_3{k}_3}\end{array}\right|\left|\begin{array}{ccc}{x}_{k_1}& x_{k_2}& x_{k_3}\\ a_{i_2{k}_1}& a_{i_2{k}_2}& a_{i_2{k}_3}\\ a_{i_3{k}_1}& a_{i_3{k}_2}& a_{i_3{k}_3}\end{array}\right|\hfill \\ \hfill =& -6+12+6=12\hfill \end{array}$$

Hence, $A_{1_3}(x,y)=\arccos \left(\frac{1}{2}\right)$.

The angle $A_{p_n}(·,·)$ has the following properties.

Proposition 3.

Let $({\ell }^p,\parallel ·,\dots ·{\parallel }_n)$ be the n-normed space. The angle $A_{p_n}$ between two nonzero vectors x and y on ${\ell }^p$ satisfies the following properties:

(a) 

If x and y are of the same direction, then $A_{p_n}(x,y)=0$; if x and y are of the opposite direction, then $A_{p_n}(x,y)=\pi$ (part of parallelism property).

(b) 

$A_{p_n}(ax,by)=A_{p_n}(x,y)$ if $ab>0$; $A_{p_n}(ax,by)=\pi -A_{p_n}(x,y)$ if $ab<0$ (homogeneity property).

(c) 

If $x_n\to x$ (in norm ${\parallel ·\parallel }_{p_n}$), then $A_{p_n}(x_n,y)\to A_{p_n}(x,y)$ (part of continuity property).

Proof. 

(a)

Let $y=kx$ for an arbitrary nonzero vector x in X and $k\in \mathbb{R}-\left\{0\right\}.$ We have

$$\begin{array}{c}\hfill A_{p_n}(x,y)=\arccos \frac{{[kx,x]}_{p_n}}{{\parallel x\parallel }_{p_n}{\parallel kx\parallel }_{p_n}}=\arccos \frac{k·{[x,x]}_{p_n}}{{\left|k\right|\parallel x\parallel }_{p_n}^2}=\arccos \frac{{k\parallel x\parallel }_{p_n}^2}{{\left|k\right|\parallel x\parallel }_{p_n}^2}.\end{array}$$

Hence, $A_{p_n}(x,y)=\arccos \left(1\right)=0$ for $k>0$ and $A_{p_n}(x,y)=\arccos (-1)=\pi$ for $k<0.$

(b)

Let $\alpha$ and $\beta \in \mathbb{R}-\left\{0\right\}.$ Observe that

$$\begin{array}{c}\hfill A_{p_n}(\alpha x,\beta y)=\arccos \frac{\alpha \beta ·{[y,x]}_{p_n}}{\left|\alpha \beta \right|(\parallel x{\parallel }_{p_n}·\parallel y{\parallel }_{p_n})}.\end{array}$$

If $\alpha \beta >0$, then $A_{p_n}(\alpha x,\beta y)=\arccos \frac{{[y,x]}_{p_n}}{{\parallel x\parallel }_{p_n}·{\parallel y\parallel }_{p_n}}=A_{p_n}(x,y).$ Likewise, if $\alpha \beta <0$, then $A_{p_n}\left(\alpha x\beta by\right)=\arccos \left(-\frac{{[y,x]}_{p_n}}{{\parallel x\parallel }_{p_n}·{\parallel y\parallel }_{p_n}}\right)=\pi -A_{p_n}(x,y)$.

(c)

If $x_k\to x$ (in norm ${\parallel ·\parallel }_{p_n}$), then

$$\begin{array}{c}\hfill |{[y,x_k-x]}_{p_n}{|\le \parallel y\parallel }_{p_n}{\parallel x_n-x\parallel }_{p_n}⟶0.\end{array}$$

Observe that ${[y,x_k-x]}_{p_n}={[y,x_k]}_{p_n}-{[y,x]}_{p_n}$. We have ${[y,x_k]}_{p_n}⟶{[y,x]}_{p_n}$. Hence,

$$\begin{array}{c}\hfill A_{{}_{p_n}}(x_n,y)\to A_{{}_{p_n}}(x,y),\end{array}$$

as desired.

 □

Remark 3.

Since the mapping ${[·,·]}_{p_n}$ in general is not commutative, the angle $A_{p_n}(·,·)$ does not satisfy the symmetry property. For example, we can see Remark 1. Likewise, the g-angle does not satisfy the continuity property.

Example 7.

For $n=1$ and $p=1$. Take $x_k=(\frac{1}{k},1,0,\cdots )$, $x=(0,1,0,\cdots )$, and $y=(1,1,0,\cdots )$ in ${\ell }^1$. We observe that $\cos A_{{}_{1_1}}(x_k,y)=0$ for every $k\in \mathbb{N}$, but $\cos A_{1_1}(x,y)\ne 0$.

2.2. $p_n$-Angle between Two Subspaces

In this section, we will discuss the $p_n$-angle between two subspaces in $({\ell }^p,\parallel ·,\dots ,·{\parallel }_p)$. In particular, we define the $p_n$-angle between a one-dimensional subspace and an m-dimensional subspace for $m\ge 1$. Using ${[·,·]}_{p_n}$ in (4), we have the Gram determinant ${\Gamma }_{p_n}$ as follows.

Definition 2.

Let $({\ell }^p,\parallel ·,\dots ,·{\parallel }_p)$ be an n-normed space and $W=span\{w_1,\cdots ,w_m\}$ fo $1\le m\le n$ is subspace of ${\ell }^p$. The Gram determinant $p_n$ of $\{w_1,\cdots ,w_m\}$, denoted by ${\Gamma }_{p_n}\{w_1,\cdots ,w_m\}$ is defined by

$$\begin{array}{c}\hfill {\Gamma }_{p_n}\{w_1,\cdots ,w_m\}:=\left|\begin{array}{ccc}{[w_1,w_1]}_{p_n}& \cdots & {[w_1,w_m]}_{p_n}\\ ⋮& \ddots & ⋮\\ [w_m,w_1{]}_{p_n}& \cdots & {[w_m,w_m]}_{p_n}\end{array}\right|.\end{array}$$

Example 8.

For $p=1$ and $n=2$. Let $\{a_1,a_2\}$ be the Schauder basis on ${\ell }^1$, that is, $a_i=\left({\delta }_{ij}\right),i=1,2,$$w_1=(1,3,0,\cdots )$ and $w_2=(3,1,0,\cdots )$ in ${\ell }^1$. We observe that $\parallel w_1{\parallel }_{1_2}=4$ and $\parallel w_2{\parallel }_{1_2}=4$. Using the semi-inner product in (4), we have

$$\begin{array}{ccc}\hfill {[w_1,w_2]}_{1_2}& =& \frac{\left({∥w_1∥}_{1_2}\right)}{2}\sum _{i=1}^2\sum _{k_1}\sum _{k_2}sgn\left|\begin{array}{cc}{w}_{1k_1}& w_{1k_2}\\ a_{i{k}_1}& a_{i{k}_2}\end{array}\right|\left|\begin{array}{cc}{w}_{2k_1}& w_{2k_2}\\ a_{i{k}_1}& a_{i{k}_2}\end{array}\right|\hfill \\ & =& 2\left(1+1+3+3\right)=16\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {[w_2,w_1]}_{1_2}& =& \frac{\left({∥w_2∥}_{1_2}\right)}{2}\sum _{i=1}^2\sum _{k_1}\sum _{k_2}sgn\left|\begin{array}{cc}{w}_{2k_1}& w_{2k_2}\\ a_{i{k}_1}& a_{i{k}_2}\end{array}\right|\left|\begin{array}{cc}{w}_{1k_1}& w_{1k_2}\\ a_{i{k}_1}& a_{i{k}_2}\end{array}\right|\hfill \\ & =& 2\left(3+3+1+1\right)=16.\hfill \end{array}$$

Hence, ${\Gamma }_{1_2}(w_1,w_2)=0$.

We have the connection between the Gram determinant $p_n$ of $\{w_1,\cdots ,w_m\}$ for $1\le m\le n$ and the linearly independence set of $\left\{w_1,\dots ,w_m\right\}$ as follows.

Theorem 2.

If ${\Gamma }_{p_n}(w_1,\cdots ,w_m)\ne 0,$ then $\left\{w_1,\cdots ,w_m\right\}$ is a linearly independent set.

Proof. 

Suppose by contradiction that $\left\{w_1,\cdots w_m\right\}$ is linearly dependent. Therefore, there is a i with $1\le i\le m$ so that $w_i$ is a linear combination of $w_1,\cdots ,w_{i-1},w_{i+1},\cdots ,w_m$. Using the properties of the determinant, we observe that the i-th column of ${\Gamma }_{p_n}$ is a linear combination of the other columns. This implies ${\Gamma }_{p_n}(w_1,\cdots ,w_m)=0$, which is a contradiction. Hence, $\left\{w_1,\cdots ,w_m\right\}$ is a linearly independent set. □

Example 9.

Suppose that $p=1$, $n=2$. Let $\{a_1,a_2\}$ be the Schauder basis on ${\ell }^1$, that is, $a_i=\left({\delta }_{ij}\right),i=1,2$ and ${\parallel ·\parallel }_{1_3}$ in Proposition 1. If $w_1=(1,0,0,0,\cdots )$ and $w_2=(0,1,0,0,\cdots )$ in ${\ell }^1$. We observe that $\parallel w_1{\parallel }_{1_2}=1$ and $\parallel w_2{\parallel }_{1_2}=1.$ As a consequence,

$$\begin{array}{c}\hfill {[w_1,w_2]}_{1_2}=\frac{\left({∥w_1∥}_{1_2}\right)}{2}\sum _{i=1}^2\sum _j\sum _{k}sgn\left|\begin{array}{cc}{w}_{1j}& w_{1k}\\ a_{ij}& a_{ik}\end{array}\right|\left|\begin{array}{cc}{w}_{2j}& w_{2k}\\ a_{ij}& a_{ik}\end{array}\right|=0.\end{array}$$

In a similar way is obtained ${[w_2,w_1]}_{1_2}=0$. As a consequence, ${\Gamma }_{1_2}(w_1,w_2)=1.$ Hence, if ${\Gamma }_{p_n}(w_1,w_2)=1\ne 0,$ then $\left\{w_1,w_2\right\}$ is a linearly independent set.

Remark 4.

The converse of the above theorem is not true. For instance, let $({\ell }^1,\parallel ·,·{\parallel }_1)$ be a 2-normed space with $a_1=(1,0,0,0,\cdots )$ and $a_2=(0,1,0,0,\cdots )$. Take $w_1=(1,3,0,\cdots )$ and $w_2=(3,1,0,\cdots )$ in ${\ell }^1$. We observe that $\parallel w_1{\parallel }_{1_2}=4$ and $\parallel w_2{\parallel }_{1_2}=4$. Using the semi-inner product in (4), we have

$$\begin{array}{ccc}\hfill {[w_1,w_2]}_{1_2}& =& \frac{\left({∥w_1∥}_{1_2}\right)}{2}\sum _{i=1}^2\sum _{k_1}\sum _{k_2}sgn\left|\begin{array}{cc}{w}_{1k_1}& w_{1k_2}\\ a_{i{k}_1}& a_{i{k}_2}\end{array}\right|\left|\begin{array}{cc}{w}_{2k_1}& w_{2k_2}\\ a_{i{k}_1}& a_{i{k}_2}\end{array}\right|\hfill \\ & =& 2\left(1+1+3+3\right)=16\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {[w_2,w_1]}_{1_2}& =& \frac{\left({∥w_2∥}_{1_2}\right)}{2}\sum _{i=1}^2\sum _{k_1}\sum _{k_2}sgn\left|\begin{array}{cc}{w}_{2k_1}& w_{2k_2}\\ a_{i{k}_1}& a_{i{k}_2}\end{array}\right|\left|\begin{array}{cc}{w}_{1k_1}& w_{1k_2}\\ a_{i{k}_1}& a_{i{k}_2}\end{array}\right|\hfill \\ & =& 2\left(3+3+1+1\right)=16.\hfill \end{array}$$

Hence, ${\Gamma }_{1_2}(w_1,w_2)=0$ but $w_1$ and $w_2$ are linearly independent.

Next, we can discuss the $p_n$-orthogonal projection and the $p_n$- orthogonal complement as follows:

Definition 3.

Let v be a vector of ${\ell }^p$ and $W=span\{w_1,\cdots ,w_m\}$ be a subspace of ${\ell }^p$ with ${\Gamma }_{p_n}(w_1,\cdots ,w_m)\ne 0$. The $p_n$-orthogonal projection of v on W, denoted by $v_{W_{p_n}}$, is defined by

$$\begin{array}{c}\hfill v_{W_{p_n}}:=-\frac{1}{{\Gamma }_{p_n}(w_1,\cdots ,w_m)}\left|\begin{array}{cccc}0& w_1& \cdots & w_m\\ {[w_1,v]}_{p_n}& {[w_1,w_1]}_{p_n}& \cdots & {[w_1,w_m]}_{p_n}\\ ⋮& ⋮& \ddots & ⋮\\ {[w_m,v]}_{p_n}& {[w_m,w_1]}_{p_n}& \cdots & {[w_m,w_m]}_{p_n}\end{array}\right|,\end{array}$$

and its $p_n$-orthogonal complement $v_{W_{p_n}}^{\perp }$ is given by

$$\begin{array}{c}\hfill v_{W_{p_n}}^{\perp }:=\frac{1}{{\Gamma }_{p_n}(w_1,\cdots ,w_m)}\left|\begin{array}{cccc}v& w_1& \cdots & w_m\\ {[w_1,v]}_{p_n}& {[w_1,w_1]}_{p_n}& \cdots & {[w_1,w_m]}_{p_n}\\ ⋮& ⋮& \ddots & ⋮\\ {[w_m,v]}_{p_n}& {[w_m,w_1]}_{p_n}& \cdots & {[w_m,w_m]}_{p_n}\end{array}\right|.\end{array}$$

Example 10.

Suppose that $p=1$, $n=2$ and $m=2$. Let $\{a_1,a_2\}$ be the Schauder basis on ${\ell }^1$, that is, $a_i=\left({\delta }_{ij}\right),i=1,2$ and ${\parallel ·\parallel }_{1_3}$ in Proposition 1. If $v=(1,1,3,0,\dots ),$$w_1=(1,0,0,0,\cdots )$ and $w_2=(0,1,0,0,\cdots )$, then ${\Gamma }_{1_2}(w_1,w_2)=1$ by Example 9. Therefore,

$$\begin{array}{ccc}\hfill v_{W_{1_2}}& =& -\frac{1}{{\Gamma }_{1_2}(w_1,w_2)}\left|\begin{array}{ccc}0& w_1& w_2\\ {[w_1,v]}_{1_2}& {[w_1,w_1]}_{1_2}& {[w_1,w_2]}_{1_2}\\ {[w_2,v]}_{1_2}& {[w_2,w_1]}_{1_2}& {[w_2,w_2]}_{1_2}\end{array}\right|\hfill \\ & =& -\left|\begin{array}{ccc}0& w_1& w_2\\ 1& 1& 0\\ 1& 0& 1\end{array}\right|=w_1+w_2=\left(1,1,0,\dots \right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill v_{W_{1_2}}^{\perp }& =& \frac{1}{{\Gamma }_{1_2}(w_1,w_2)}\left|\begin{array}{ccc}v& w_1& w_2\\ {[w_1,v]}_{1_2}& {[w_1,w_1]}_{1_2}& {[w_1,w_2]}_{1_2}\\ {[w_2,v]}_{1_2}& {[w_2,w_1]}_{1_2}& {[w_2,w_2]}_{1_2}\end{array}\right|\hfill \\ & =& \left|\begin{array}{ccc}v& w_1& w_2\\ 1& 1& 0\\ 1& 0& 1\end{array}\right|=v-w_1-w_2=\left(0,0,3,0,\dots \right).\hfill \end{array}$$

Using the properties of the determinant and the semi-inner product in (4), we obtain

$$\begin{array}{c}\hfill {[w_i,v_{W_{p_n}}^{\perp }]}_{p_n}=\frac{1}{{\Gamma }_{p_n}(w_1,\cdots ,w_m)}\left|\begin{array}{cccc}{[w_i,v]}_{p_n}& {[w_i,w_1]}_{p_n}& \cdots & {[w_i,w_m]}_{p_n}\\ {[w_1,v]}_{p_n}& {[w_1,w_1]}_{p_n}& \cdots & {[w_1,w_m]}_{p_n}\\ ⋮& ⋮& \ddots & ⋮\\ {[w_m,v]}_{p_n}& {[w_m,w_1]}_{p_n}& \cdots & {[w_m,w_m]}_{p_n}\end{array}\right|=0.\end{array}$$

Hence, $w_i{\perp }_{p_n}{v}_{W_{p_n}}^{\perp }$ for every $i=1,\cdots ,n$.

The definition of the $p_n$-angle between $V=\mathrm{span}\left\{v\right\}$ and $W=\mathrm{span}\{w_1,\cdots ,w_m\}$ of $({\ell }^p,\parallel ·,\dots ,·{\parallel }_p)$ is as follows.

Definition 4.

Let $({\ell }^p,{∥·,\dots ,·∥}_p)$ be an n-normed space for $1\le p<\infty$. If $V=span\left\{v\right\}$ and $W=span\{w_1,\cdots ,w_m\}$ are subspaces of ${\ell }^p$ with ${\Gamma }_{p_n}(w_1,\cdots ,w_m)\ne 0$ dan $m\ge 1$ then the $p_n$-angle between subspaces V and W is denoted by $A_{p_n}(V,W)$ with

$${\cos }^2{A}_{p_n}(V,W)=\frac{{[v_{W_{p_n}},v]}_{p_n}^2}{{\left({∥v∥}_{p_n}\right)}^2{\left({∥v_{W_{p_n}}∥}_{p_n}\right)}^2}.$$

(6)

where $v_{W_{p_n}}$ denotes the $p_n$-orthogonal projection of v on W.

Example 11.

Suppose that $p=1$, $n=2$ and $m=2$ in Definition 4 Let $\{a_1,a_2\}$ be the Schauder basis on ${\ell }^1$, that is, $a_i=\left({\delta }_{ij}\right),i=1,2$ and ${\parallel ·\parallel }_{1_3}$ in Proposition 1. If $V=span\left\{v\right\}$ and $W=span\{w_1,w_2\}$ with $v=(1,1,3,0,\dots ),$$w_1=(1,0,0,0,\cdots )$ and $w_2=(0,1,0,0,\cdots )$ in ${\ell }^1$, we observe that

$$\begin{array}{cc}\hfill {\parallel v\parallel }_{1_2}& =\parallel v,a_1,a_2{\parallel }_1+{\parallel v,a_1,a_2\parallel }_1\hfill \\ \hfill & =\frac{1}{2}\sum _{k_1}\sum _{k_2}\left(abs\left|\begin{array}{cc}{v}_{k_1}& v_{k_2}\\ a_{1k_1}& a_{1k_2}\end{array}\right|\right)+\frac{1}{2}\sum _{k_1}\sum _{k_2}\left(abs\left|\begin{array}{cc}{v}_{k_1}& v_{k_2}\\ a_{2k_1}& a_{2k_2}\end{array}\right|\right)\hfill \\ \hfill & =4+4=8.\hfill \end{array}$$

By using Example 9, we have ${\Gamma }_{1_2}(w_1,w_2)=1.$ Therefore,

$$\begin{array}{ccc}\hfill v_{W_{1_2}}& =& -\frac{1}{{\Gamma }_{1_2}(w_1,w_2)}\left|\begin{array}{ccc}0& w_1& w_2\\ {[w_1,v]}_{1_2}& {[w_1,w_1]}_{1_2}& {[w_1,w_2]}_{1_2}\\ {[w_2,v]}_{1_2}& {[w_2,w_1]}_{1_2}& {[w_2,w_2]}_{1_2}\end{array}\right|\hfill \\ & =& -\left|\begin{array}{ccc}0& w_1& w_2\\ 1& 1& 0\\ 1& 0& 1\end{array}\right|=w_1+w_2=\left(1,1,0,\dots \right).\hfill \end{array}$$

Moreover, we observe that ${∥v_{W_{1_2}}∥}_{1_2}=2$ and ${[v_{W_{1_2}},v]}_{1_2}=4$. Hence,

$$\begin{array}{c}\hfill {\cos }^2{A}_{1_2}(V,W)=\frac{{[v_{W_{1_2}},v]}_{1_2}^2}{{\left({∥v∥}_{1_2}\right)}^2{\left({∥v_{W_{1_2}}∥}_{1_2}\right)}^2}=\frac{1}{16}.\end{array}$$

Remark 5.

If $W=span\left\{w\right\}$, then the $p_n$-orthogonal projection of v on W is

$$v_{W_{p_n}}=\frac{{[w,v]}_{p_n}}{{\parallel w\parallel }_{p_n}^2}w.$$

Using this $p_n$-orthogonal projection, the properties of the semi-inner product $p_n$ and Definition 4, we have the $p_n$-angle between subspaces V and W, that is,

$${\cos }^2{A}_{p_n}(V,W)=\frac{{[w,v]}_{p_n}^2}{{\left({∥v∥}_{p_n}\right)}^2{\left({∥w∥}_{p_n}\right)}^2}.$$

Hence, we can see that the $p_n$-angle between the two vectors is also the $p_n$-angle between these two vectors in the subspace spanned by them. Next, by using Definition 3, we have $v=v_{W_{p_n}}+v_{W_{p_n}}^{\perp }$. Therefore, Definition 4 may be rewritten as

$$\begin{array}{cc}\hfill {\cos }^2{A}_{p_n}(V,W)& =\frac{{[v_{W_{p_n}},v_{W_{p_n}}+v_{W_{p_n}}^{\perp }]}_{p_n}^2}{{\left({∥v∥}_{p_n}\right)}^2{\left({∥v_{W_{p_n}}∥}_{p_n}\right)}^2}=\frac{{\left({[v_{W_{p_n}},v_{W_{p_n}}]}_{p_n}+{[v_{W_{p_n}},v_{W_{p_n}}+v_{W_{p_n}}^{\perp }]}_{p_n}\right)}^2}{{\left({∥v∥}_{p_n}\right)}^2{\left({∥v_{W_{p_n}}∥}_{p_n}\right)}^2}\hfill \\ \hfill & =\frac{{[v_{W_{p_n}},v_{W_{p_n}}]}_{p_n}^2}{{\left({∥v∥}_{p_n}\right)}^2{\left({∥v_{W_{p_n}}∥}_{p_n}\right)}^2}=\frac{(\parallel v_{W_{p_n}}{{\parallel }_{p_n})}^2}{{\left({∥v∥}_{p_n}\right)}^2}.\hfill \end{array}$$

This tells us that the value of $\cos A_{p_n}(V,W)$ is equal to the ratio between the norm ${\parallel ·\parallel }_{p_n}$ of the $p_n$-orthogonal projection of v on W and the norm ${\parallel ·\parallel }_{p_n}$ of v.

3. Further Results

Let X be a measured space with at least n disjoint subsets of positive measure. Our results also extend the n-normed space $(L^p\left(X\right),\parallel ·,\dots ,·{\parallel }_{L^p})$ for $1\le p<\infty$ with a n-norm that was defined by Gunawan in [17]

$$\parallel f_1,\dots ,f_n{\parallel }_{L^p}=\frac{1}{n!}\underset{X}{\int }\dots \underset{X}{\int }{\left(\mathrm{abs}\left|\begin{array}{ccc}{f}_1\left(x_1\right)& \cdots & f_n\left(x_n\right)\\ ⋮& \ddots & ⋮\\ f_n\left(x_1\right)& \cdots & f_n\left(x_n\right)\end{array}\right|\right)}^{p}d{x}_1\dots d{x}_2.$$

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Next, Ekariani et al. [18] defines a norm on $L^p\left(X\right)$ by

$${\parallel f\parallel }_{L^{p_n}}:={\left[\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}{\parallel f,a_{i_2},\dots ,a_{i_n}\parallel }_{L^p}\right]}^{\frac{1}{p}},$$

where $\{a_1,\dots ,a_n\}$ is a linearly independent set in $L^p\left(X\right)$. Using the norm ${\parallel ·\parallel }_{L^{p_n}}$ with $a_i=\left(a_{ij}\right)$ for $i=1,\dots ,n$, we define a mapping ${[·,·]}_{L^{p_n}}$ with $1\le p<\infty$ by

$$\begin{array}{cc}\hfill {[f,g]}_{L^{p_n}}& =\left[\frac{{\left({∥f∥}_{L^{p_n}}\right)}^{2-p}}{n!}\sum _{\{i_2,\dots ,i_n\}\subset \{1,\dots ,n\}}\underset{X}{\int }\dots \underset{X}{\int }{\left(\mathrm{abs}\left|\begin{array}{ccc}f\left(x_1\right)& \cdots & f\left(x_n\right)\\ a_{i_2}\left(x_1\right)& \cdots & a_{i_2}\left(x_n\right)\\ ⋮& \ddots & ⋮\\ a_{i_n}\left(x_1\right)& \cdots & a_{i_n}\left(x_n\right)\end{array}\right|\right)}^{p-1}\times \right.\hfill \\ \hfill & \times \left.\mathrm{sgn}\left|\begin{array}{ccc}f\left(x_1\right)& \cdots & f\left(x_n\right)\\ a_{i_2}\left(x_1\right)& \cdots & a_{i_2}\left(x_n\right)\\ ⋮& \ddots & ⋮\\ a_{i_n}\left(x_1\right)& \cdots & a_{i_n}\left(x_n\right)\end{array}\right|\left|\begin{array}{ccc}g\left(x_1\right)& \cdots & g\left(x_n\right)\\ a_{i_2}\left(x_1\right)& \cdots & a_{i_2}\left(x_n\right)\\ ⋮& \ddots & ⋮\\ a_{i_n}\left(x_1\right)& \cdots & a_{i_n}\left(x_n\right)\end{array}\right|d{x}_1\dots d{x}_2\right].\hfill \end{array}$$

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and check ${[f,g]}_{L^{p_n}}$ defines a semi-inner product on $L^p\left(X\right)$. Next, the results are analogous to section main results.

4. Conclusions

In this article, we have introduced the semi-inner product in an n-normed space $({\ell }^p,\parallel ·,\dots ,·{\parallel }_p)$ with $1\le p<\infty$. We have introduced the $p_n$-orthogonal and the $p_n$-angle between two vectors. We have proven their properties. Moreover, we have introduced the notion of $p_n$-angle between one-dimensional subspaces and arbitrary-dimensional subspaces in the n-normed space $({\ell }^p,\parallel ·,\cdots ,·{\parallel }_p)$.