Relationship between Generalized Orthogonality and Gâteaux Derivative

Abstract

This paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of x in the y direction is zero when the norm is Gâteaux differentiable in the y direction at x and x and y satisfy certain generalized orthogonality conditions. A case where x and y are approximately orthogonal is also analyzed and the value range of the Gâteaux derivative in this case is given. Moreover, two concepts are introduced: the angle between vectors in normed linear space and the ${\perp }_{\Delta }$ coordinate system in a smooth Minkowski plane. Relevant examples are given at the end of the paper.

1. Introduction

Generalized orthogonality plays an important role in the study of normed linear spaces. It can generalize important conclusions from inner product spaces to normed linear spaces. Many scholars have extended different forms of orthogonality in inner product spaces to normed linear spaces [1,2,3,4,5,6,7,8,9,10,11,12], such as Birkhoff orthogonality [1] and Roberts orthogonality [2]. With the in-depth study of generalized orthogonality theory, the concept of approximate orthogonality has also been proposed [13,14,15]. Since the norm is a non-negative function, it is of great importance to study the Gâteaux derivative of the norm. Abatzoglou T. J. introduced the concept of the Gâteaux derivative of a norm in a real normed linear space in 1979 (see [16]). In 2005, Kečkié D. J. gave a definition of the Gâteaux derivative of the B(H)-norm of all bounded linear operators in Hilbert spaces and characterized the Birkhoff orthogonality by this result (see [17]). In 2009, Alsina C. and Sikorska J. presented the characteristics of the norm derivatives in the inner product space (see [18]). In 2023, Wójcik P. and Zamani A. established the relation between the $C^{*}$-module and the norm derivative and the orthogonal in Hilbert spaces (see [19]). With the in-depth study of generalized orthogonality, this paper tries to combine various generalized orthogonality, approximate orthogonality, and the Gâteaux derivative in normed linear space and finds that some interesting conclusions can be drawn.

2. Preliminaries

Let $\left(X,∥·∥\right)$ be a normed linear space, and $X^{*}$ is the dual space of X. The sets $B_X:=\{x\in X|∥x∥\le 1\}$ and $S_X:=\{x\in X|∥x∥=1\}$ are called the unit ball and unit sphere. A real finite-dimensional normed linear space is called a Minkowski space, and a two-dimensional Minkowski space is called a Minkowski plane. A normed linear space X is called an inner product space if the norm $∥·∥$ on X is induced by an inner product, i.e., if there exists an inner product $⟨·|·⟩$ on X such that $∥x∥=\sqrt{⟨x|x⟩}$, $\forall x\in X$. If two vectors x and y satisfy $⟨x|y⟩=0$, let us call x orthogonal to y, and we write $x\perp y$. However, no such binary relation exists in the normed linear space. Thus, it is necessary to extend the orthogonal relation in the inner product space to the normed linear space and call it generalized orthogonality. Although the various forms of generalized orthogonality are different in normed linear spaces, they are the same in inner product spaces. The following describes the generalized orthogonal relations used in this paper. Let $\mathbb{R}$ be the set of all real numbers, $x,y\in X$, and x is said to be Birkhoff orthogonal [1] to y (denoted by $x{\perp }_{B}y$ ) if

$$∥x+\lambda y∥\ge ∥x∥,\forall \lambda \in \mathbb{R};$$

x is said to be Robert orthogonal [2] to y (denoted by $x{\perp }_{R}y$ ) if

$$∥x+\lambda y∥=∥x-\lambda y∥,\forall \lambda \in \mathbb{R};$$

x is said to be isosceles orthogonal [10] to y (denoted by $x{\perp }_{I}y$ ) if

$$∥x+y∥=∥x-y∥;$$

x is said to be Pythagorean orthogonal [10] to y (denoted by $x{\perp }_{P}y$ ) if

$${∥x-y∥}^2={∥x∥}^2+{∥y∥}^2;$$

x is said to be $\lambda p$ orthogonal [12] to y (denoted by $x{\perp }_{\lambda p}y$ ) if

$${∥x+\lambda y∥}^2={∥x∥}^2+{\lambda }^2{∥y∥}^2,\forall \lambda \in \mathbb{R},\lambda \ne 0.$$

Compared with generalized orthogonality, approximate orthogonality in normed linear spaces has more relaxed conditions. The following is the concept of approximate orthogonality used in this paper. x is said to be $\epsilon$-Birkhoff orthogonal [13] to y (denoted by $x{\perp }_B^{\epsilon }y$ ) if

$${∥x+\lambda y∥}^2\ge {∥x∥}^2-2\epsilon ∥x∥∥\lambda y∥,\forall \lambda \in \mathbb{R},\epsilon \in \left[0,1\right);$$

x is said to be Birkhoff-$\epsilon$ orthogonal [14] to y (denoted by $x{\perp }_{B\epsilon }y$) if

$$∥x+\lambda y∥\ge ∥x∥-\epsilon ∥\lambda y∥,\forall \lambda \in \mathbb{R},\epsilon \in \left[0,1\right);$$

x is said to be $\epsilon$-Robert orthogonal [15] to y (denoted by $x{\perp }_R^{\epsilon }y$) if

$$|{∥x+\lambda y∥}^2-{∥x-\lambda y∥}^2|\le 4\epsilon ∥x∥∥\lambda y∥,\forall \lambda \in \mathbb{R},\epsilon \in \left[0,1\right).$$

The following is the definition of the semi-inner product (S.I.P.) (see [20]), which is a generalization of the inner product and is more widely used in the inner product space. Let Y be a real linear space. Each pairing of points $x,y\in Y$ is a real number $\left[x|y\right]$. The real number $\left[x|y\right]$ satisfies these properties:

(i)

$\left[x|x\right]>0$, $\forall x\ne 0$;

(ii)

$\left[x+y|z\right]=\left[x|z\right]+\left[y|z\right]$, $\forall x,y,z\in Y$;

(iii)

$\left[\lambda x|y\right]=\lambda \left[x|y\right]$, $\forall x,y\in Y$, $\forall \lambda \in$$\mathbb{R}$;

(iv)

$\left[x|\alpha y\right]=\alpha \left[x|y\right]$, $\forall \alpha \in$$\mathbb{R}$;

(v)

$|\left[x|y\right]{|}^2\le \left[x|x\right]\left[y|y\right]$, $\forall x,y\in Y$.

Then, Y is said to be an S.I.P. space, $\left[·|·\right]$ is said to be an S.I.P. of Y. If $\left[·|·\right]$ is an S.I.P. on X, and $∥x∥=\sqrt{⟨x|x⟩}$ holds for $\forall x\in X$; then, $\left[·|·\right]$ is said to be a generating norm S.I.P.

Let X be a normed linear space, $S_X$ be the unit sphere of X, $x\in S_X$, $y\in X$,

$${\displaystyle G_{+}\left(x,y\right)=\underset{\lambda \to 0^{+}}{lim}\frac{∥x+\lambda y∥-∥x∥}{\lambda },}$$

$${\displaystyle G_{-}\left(x,y\right)=\underset{\lambda \to 0^{-}}{lim}\frac{∥x+\lambda y∥-∥x∥}{\lambda }.}$$

We refer to $G_{+}\left(x,y\right)$ and $G_{-}\left(x,y\right)$ as the right and left Gâteaux derivatives of the norm $\parallel ·\parallel$ at x in the direction of y (see [21]). If

$$G_{+}\left(x,y\right)=G_{-}\left(x,y\right)=G\left(x,y\right)$$

is true, we consider the norm $\parallel ·\parallel$ to be differentiable in the y direction at x, where $G\left(x,y\right)$ represents the derivative of the norm $\parallel ·\parallel$ in the y direction at x. If the norm $\parallel ·\parallel$ is differentiable at x in any direction y, we say that the norm is differentiable at x. Regarding orthogonality related to S.I.P., we say that x is Lumer orthogonal [13] to y (denoted by $x{\perp }_{L}y$) if

$$\left[y|x\right]=0.$$

Let $x\in S_X$, and let there exist a unique bounded linear functional $x^{*}\in S_{X^{*}}$ at x such that $x^{*}\left(x\right)=1$ holds; then, x is said to be a smooth point of $B_X$ [22]. If each point of $S_X$ is smooth, then X is said to be smooth.

3. Results

3.1. Relationship between Orthogonality, Approximate Orthogonality, and Gâteaux Derivative

To demonstrate the validity of the theorems presented in this section, we must employ the two lemmas established by prior scholars.

Lemma 1. 

Let X be a normed linear space, $x,y\in X$. ${\left\{x_n\right\}}_{n=1}^{\infty }$, ${\left\{y_n\right\}}_{n=1}^{\infty }$ be two columns of vectors in X, $\underset{n\to \infty }{lim}{x}_n=x$, and $\underset{n\to \infty }{lim}{y}_n=y$. For any natural number $n>0$, if $x_n{\perp }_P{y}_n$ is true, then $x{\perp }_{P}y$.

Proof of Lemma 1. 

The proof that $x{\perp }_{P}y$ is true is simply the proof that

$${∥x-y∥}^2={∥x∥}^2+{∥y∥}^2$$

is true. The equation

$${∥x_n-y_n∥}^2={∥x_n∥}^2+{∥y_n∥}^2$$

obtained by $x_n{\perp }_P{y}_n$ is true, i.e.,

$$\begin{array}{cc}\hfill {∥x-y∥}^2& =\underset{n\to \infty }{lim}{∥x_n-y_n∥}^2\hfill \\ & =\underset{n\to \infty }{lim}{∥x_n∥}^2+\underset{n\to \infty }{lim}{∥y_n∥}^2\hfill \\ & ={∥x∥}^2+{∥y∥}^2,\hfill \end{array}$$

then, $x{\perp }_{P}y$ is proven. □

Lemma 2. 

Let X be a normed linear space, $x,y\in X$. ${\left\{x_n\right\}}_{n=1}^{\infty }$, ${\left\{y_n\right\}}_{n=1}^{\infty }$ be two columns of vectors in X, $\underset{n\to \infty }{lim}{x}_n=x$, $\underset{n\to \infty }{lim}{x}_n=x$. For any natural number $n>0$, if $x_n{\perp }_I{y}_n$ is true, then $x{\perp }_{I}y$.

Proof of Lemma 2. 

The proof of $x{\perp }_{I}y$ is simply the proof that the equation

$$∥x-y∥=∥x+y∥$$

is true. The equation

$$∥x_n-y_n∥=∥x_n+y_n∥$$

obtained by $x_n{\perp }_I{y}_n$ is true, i.e.,

$$\begin{array}{cc}\hfill ∥x-y∥& =\underset{n\to \infty }{lim}∥x_n-y_n∥\hfill \\ \hfill & =\underset{n\to \infty }{lim}∥x_n+y_n∥=∥x+y∥,\hfill \end{array}$$

then $x{\perp }_{I}y$ is proven. □

Theorem 1. 

Let X be a normed linear space, $x\in S_X$, $y\in X$. If the norm $∥·∥$ at x is Gâteaux differentiable along the direction y, then $G\left(x,y\right)=0$, and any of the following conditions are sufficient: (i) $x{\perp }_{B}y$; (ii) $x{\perp }_{R}y$; (iii) $x{\perp }_{P}y$; (iv) $x{\perp }_{I}y$; (v) there exists an S.I.P. of the generating norm on X such that $x{\perp }_{L}y$.

Proof of Theorem 1. 

(i)

Since $x{\perp }_{B}y$, it follows that $∥x+\lambda y∥\ge ∥x∥$, which implies $∥x+\lambda y∥-∥x∥\ge 0$. Consequently,

$${\displaystyle \underset{\lambda \to 0^{+}}{lim}\frac{∥x+\lambda y∥-∥x∥}{\lambda }\ge 0,}$$

that is, $G_{+}\left(x,y\right)\ge 0$. Similarly,

$${\displaystyle \underset{\lambda \to 0^{-}}{lim}\frac{∥x+\lambda y∥-∥x∥}{\lambda }\le 0,}$$

such that is $G_{-}\left(x,y\right)\le 0$. Since the norm $∥·∥$ is Gâteaux differentiable at x along the y direction, we have

$$G_{-}\left(x,y\right)=G_{+}\left(x,y\right),$$

hence $G\left(x,y\right)=0$.

(ii)

Due to $x{\perp }_{R}y$, the equation $∥x+\lambda y∥=∥x-\lambda y∥$ holds, thus

$$∥x+\lambda y∥-∥x∥=∥x-\lambda y∥-∥x∥.$$

Then,

$$\begin{array}{cc}\hfill G_{+}\left(x,y\right)& {\displaystyle =\underset{\lambda \to 0^{+}}{lim}\frac{∥x+\lambda y∥-∥x∥}{\lambda }}\hfill \\ & {\displaystyle =-\underset{\lambda \to 0^{+}}{lim}\frac{∥x-\lambda y∥-∥x∥}{-\lambda }.}\hfill \end{array}$$

Let ${\lambda }_1=-\lambda$; then, we can obtain

$$\begin{array}{cc}\hfill G_{+}\left(x,y\right)& {\displaystyle =-\underset{{\lambda }_1\to 0^{-}}{lim}\frac{∥x-{\lambda }_{1}y∥-∥x∥}{{\lambda }_1}}\hfill \\ \hfill & =-G_{-}\left(x,y\right).\hfill \end{array}$$

As the norm $∥·∥$ is Gâteaux differentiable at x along y direction, then

$$G\left(x,y\right)=G_{+}\left(x,y\right)=G_{-}\left(x,y\right)=0.$$

(iii)

The expression $∥x+\lambda y∥$ is needed in the process of solving the derivative of Gâteaux at x along the y direction of the norm $∥·∥$, but $∥x+\lambda y∥$ does not exist in the definition of Pythagorean orthogonality. To calculate the derivative of Gâteaux of norm $∥·∥$ along the y direction at x, we need to construct $∥x+{\lambda }_{n}y∥$ and it needs to satisfy $\underset{n\to \infty }{lim}∥x+{\lambda }_{n}y∥=∥x∥$. The methods of construction are shown in the following.

We need to construct two columns of vectors ${\left\{x_n\right\}}_{n=1}^{\infty }$, ${\left\{y_n\right\}}_{n=1}^{\infty }$ and $x_n{\perp }_P{y}_n$.

Let $x_n=x+{\lambda }_{n}y$, $y_n=y+{\lambda }_{n}y$, and $x_n{\perp }_P{y}_n$, where ${\lambda }_n$ is a monotonically decreasing sequence, ${\lambda }_1>0$, and $\underset{n\to \infty }{lim}{\lambda }_n=0$. Subsequently, we can calculate

$$\underset{n\to \infty }{lim}{x}_n=x+\underset{n\to \infty }{lim}{\lambda }_{n}y=x,$$

$$\underset{n\to \infty }{lim}{y}_n=y+\underset{n\to \infty }{lim}{\lambda }_{n}y=y,$$

and

$$\underset{n\to \infty }{lim}{∥x_n-y_n∥}^2=\underset{n\to \infty }{lim}{∥x_n∥}^2+\underset{n\to \infty }{lim}{∥y_n∥}^2.$$

From the above, we can obtain $\underset{n\to \infty }{lim}∥x+{\lambda }_{n}y∥=∥x∥$.

Then, under the condition of Pythagorean orthogonality, we begin to solve for the value of the Gâteaux derivative of the norm at x in the y direction.

From Lemma 1, we know that if $x_n{\perp }_P{y}_n$, $x{\perp }_{P}y$ is true, i.e., the equation

$${∥x-y∥}^2={∥x∥}^2+{∥y∥}^2$$

is true. Then, we can deduce

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{∥x+{\lambda }_{n}y∥}^2+\underset{n\to \infty }{lim}{∥y+{\lambda }_{n}y∥}^2& =\underset{n\to \infty }{lim}{∥x_n∥}^2+\underset{n\to \infty }{lim}{∥y_n∥}^2\hfill \\ \hfill & =\underset{n\to \infty }{lim}{∥x_n-y_n∥}^2\hfill \\ \hfill & =\underset{n\to \infty }{lim}{∥x+{\lambda }_{n}y-y-{\lambda }_{n}y∥}^2={∥x-y∥}^2\hfill \end{array}$$

from it, i.e.,

$$\underset{n\to \infty }{lim}{∥x+{\lambda }_{n}y∥}^2={∥x-y∥}^2-\underset{n\to \infty }{lim}{∥y+{\lambda }_{n}y∥}^2.$$

Consequently,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{∥x+{\lambda }_{n}y∥}^2-{∥x∥}^2& ={∥x-y∥}^2-\underset{n\to \infty }{lim}{∥y+{\lambda }_{n}y∥}^2-{∥x∥}^2\hfill \\ & ={∥y∥}^2-\underset{n\to \infty }{lim}{∥y+{\lambda }_{n}y∥}^2\hfill \\ & \le {∥y∥}^2-{∥y∥}^2+2\underset{n\to \infty }{lim}∥y∥∥{\lambda }_{n}y∥-\underset{n\to \infty }{lim}{∥{\lambda }_{n}y∥}^2=0.\hfill \end{array}$$

Therefore,

$${\displaystyle ∥x∥G_{+}\left(x,y\right)=\underset{n\to \infty }{lim}\frac{{∥x+{\lambda }_{n}y∥}^2-{∥x∥}^2}{2{\lambda }_n}\le 0,}$$

that is, $G_{+}\left(x,y\right)\le 0$. Likewise, let $x_n=x+{\lambda }_{n}y$, $y_n=y+{\lambda }_{n}y$ but ${\lambda }_n$ is a monotonically increasing sequence, ${\lambda }_1<0$ and $\underset{n\to \infty }{lim}{\lambda }_n=0$. Then, $\underset{n\to \infty }{lim}{x}_n=x$, $\underset{n\to \infty }{lim}{y}_n=y$.

Hence, $G_{-}\left(x,y\right)\ge 0$. As the norm $∥·∥$ is Gâteaux differentiable at x along the y direction, then

$$G\left(x,y\right)=G_{+}\left(x,y\right)=G_{-}\left(x,y\right)=0.$$

(iv)

The expression $∥x+\lambda y∥$ is needed in the process of solving the derivative of the Gâteaux at x along the y direction of the norm $∥·∥$, but $∥x+\lambda y∥$ does not exist in the definition of isosceles orthogonality. To calculate the derivative of the Gâteaux of the norm $∥·∥$ along the y direction at x, we need to construct $∥x+{\lambda }_{n}y∥$ and it needs to satisfy $\underset{n\to \infty }{lim}∥x+{\lambda }_{n}y∥=∥x∥$. The methods of construction are shown in the following.

We need to construct two columns of vectors ${\left\{x_n\right\}}_{n=1}^{\infty }$ and ${\left\{y_n\right\}}_{n=1}^{\infty }$ and $x_n{\perp }_I{y}_n$.

Let ${\displaystyle x_n=x+\frac{{\lambda }_n}{2}y}$, ${\displaystyle y_n=y-\frac{{\lambda }_n}{2}y}$ and $x_n{\perp }_I{y}_n$, where ${\lambda }_n$ is a monotonically decreasing sequence, ${\lambda }_1>0$ and $\underset{n\to \infty }{lim}{\lambda }_n=0$. Subsequently, we can calculate

$${\displaystyle \underset{n\to \infty }{lim}{x}_n=x+\underset{n\to \infty }{lim}\frac{{\lambda }_n}{2}y=x}$$

$${\displaystyle \underset{n\to \infty }{lim}{y}_n=y-\underset{n\to \infty }{lim}\frac{{\lambda }_n}{2}y=y}$$

and

$$\underset{n\to \infty }{lim}∥x_n-y_n∥=\underset{n\to \infty }{lim}∥x_n+y_n∥.$$

From the above, we can obtain $\underset{n\to \infty }{lim}∥x+{\lambda }_{n}y∥=∥x∥$.

Then, under the condition of isosceles orthogonality, we begin to solve for the value of the Gâteaux derivative of the norm at x in the y direction.

From Lemma 2, we know that if $x_n{\perp }_I{y}_n$, $x{\perp }_{I}y$ is true, i.e., the equation

$$∥x-y∥=∥x+y∥$$

is true. Then, we can deduce

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}∥x+\frac{{\lambda }_n}{2}y+y-\frac{{\lambda }_n}{2}y∥& =\underset{n\to \infty }{lim}∥x_n+y_n∥=\underset{n\to \infty }{lim}∥x_n-y_n∥\hfill \\ \hfill & =\underset{n\to \infty }{lim}∥x+\frac{{\lambda }_n}{2}y-y+\frac{{\lambda }_n}{2}y∥\hfill \\ \hfill & =\underset{n\to \infty }{lim}∥x-y+{\lambda }_{n}y∥=∥x-y∥\hfill \end{array}$$

from it, i.e.,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}∥x+{\lambda }_{n}y∥-∥x∥& \ge ∥x-y∥-∥x∥-∥y∥\hfill \\ \hfill & \ge ∥x∥+∥y∥-∥x∥-∥y∥=0.\hfill \end{array}$$

Consequently,

$${\displaystyle G_{+}\left(x,y\right)=\underset{n\to \infty }{lim}\frac{∥x+{\lambda }_{n}y∥-∥x∥}{\lambda }\ge 0,}$$

that is, $G_{+}\left(x,y\right)\ge 0$. Similarly, let

$${\displaystyle x_n=x+\frac{{\lambda }_n}{2}y,y_n=y-\frac{{\lambda }_n}{2}y,}$$

but ${\lambda }_n$ is a monotonically increasing sequence, ${\lambda }_1<0$, and $\underset{n\to \infty }{lim}{\lambda }_n=0$. Then, $\underset{n\to \infty }{lim}{x}_n=x$, $\underset{n\to \infty }{lim}{y}_n=y$. Thus, $G_{-}\left(x,y\right)\le 0$. As the norm $G_{-}\left(x,y\right)\le 0$ is Gâteaux differentiable at x along the y direction, then

$$G\left(x,y\right)=G_{+}\left(x,y\right)=G_{-}\left(x,y\right)=0.$$

(v)

There exists an S.I.P. on X that generates the norm $x{\perp }_{L}y$, so

$$\left[{\displaystyle \frac{y}{∥y∥},\frac{x}{∥x∥}}\right]=0,$$

therefore,

$$G_{-}\left(x,y\right)\le \left[{\displaystyle \frac{y}{∥y∥},\frac{x}{∥x∥}}\right]\le G_{+}\left(x,y\right).$$

Moreover, as the norm $∥·∥$ is Gâteaux differentiable at x along the y direction, we have

$$G_{-}\left(x,y\right)=G_{+}\left(x,y\right)=0.$$

In summary, $G\left(x,y\right)=0$.

□

Theorem 2. 

Let X be a normed linear space, $x\in S_X$, $y\in X$. If $x{\perp }_{\lambda P}y$, then the norm $∥·∥$ is Gâteaux differentiable in the y direction at x and $G\left(x,y\right)=0$.

Proof of Theorem 2. 

Since $x{\perp }_{\lambda P}y$, it follows that

$${∥x+\lambda y∥}^2={∥x∥}^2+{\lambda }^2{∥y∥}^2,\lambda \ne 0.$$

When $y=0$, it is obviously true. Next, we discuss the case of $y\ne 0$.

$${\displaystyle ∥x∥G_{+}\left(x,y\right)=\underset{\lambda \to 0^{+}}{lim}\frac{{∥x+\lambda y∥}^2-{∥x∥}^2}{2\lambda }=\underset{\lambda \to 0^{+}}{lim}\frac{{\lambda }^2{∥y∥}^2}{2\lambda }=0,}$$

then $G_{+}\left(x,y\right)=0$. Likewise,

$${\displaystyle ∥x∥G_{-}\left(x,y\right)=\underset{\lambda \to 0^{-}}{lim}\frac{{∥x+\lambda y∥}^2-{∥x∥}^2}{2\lambda }=\underset{\lambda \to 0^{-}}{lim}\frac{{\lambda }^2{∥y∥}^2}{2\lambda }=0,}$$

then $G_{-}\left(x,y\right)=0$. From the above, we obtain

$$G\left(x,y\right)=G_{+}\left(x,y\right)=G_{-}\left(x,y\right)=0.$$

Then, the norm $∥·∥$ is Gâteaux differentiable at x along the y direction and $G\left(x,y\right)=0$. □

Theorem 3. 

Let X be a normed linear space, $x,y\in S_X$. If $x{\perp }_B^{\epsilon }y$ and the norm $∥·∥$ at x is Gâteaux differentiable along the direction y, then $G\left(x,y\right)\in \left[-\epsilon ,\epsilon \right]$, $\epsilon \in \left[0,1\right)$.

Proof of Theorem 3. 

According to $x{\perp }_B^{\epsilon }y$, we have

$${∥x+\lambda y∥}^2\ge {∥x∥}^2-2\epsilon ∥x∥∥\lambda y∥,\epsilon \in \left[0,1\right).$$

When $\lambda >0$, we have

$${\displaystyle \underset{\lambda \to 0^{+}}{lim}\frac{{∥x+\lambda y∥}^2-{∥x∥}^2}{2\lambda }\ge {\displaystyle \frac{-\epsilon ∥x∥∥\lambda y∥}{\lambda }},}$$

that is,

$$∥x∥G_{+}\left(x,y\right)\ge -\epsilon ∥x∥∥y∥.$$

Then,

$$G_{+}\left(x,y\right)\ge -\epsilon ∥y∥.$$

Similarly, when $\lambda <0$, we have

$${\displaystyle \underset{\lambda \to 0^{-}}{lim}\frac{{∥x+\lambda y∥}^2-{∥x∥}^2}{2\lambda }\le \frac{-\epsilon ∥x∥∥\lambda y∥}{\lambda },}$$

that is,

$$∥x∥G_{-}\left(x,y\right)\le \epsilon ∥x∥∥y∥.$$

Thus,

$$G_{-}\left(x,y\right)\le \epsilon ∥y∥.$$

Since the norm $∥·∥$ is Gâteaux differentiable at x along the y direction, then

$$-\epsilon ∥y∥\le G\left(x,y\right)\le \epsilon ∥y∥.$$

Due to $y\in S_X$, then $G\left(x,y\right)\in \left[-\epsilon ,\epsilon \right]$, where $\epsilon \in \left(0,1\right]$. □

Theorem 4. 

Let X be a normed linear space, $x,y\in S_X$, and $x{\perp }_{B\epsilon }y$. If the norm $∥·∥$ is Gâteaux derivable in the y direction at x, then $G\left(x,y\right)\in \left[-\epsilon ,\epsilon \right]$.

Proof of Theorem 4. 

From $x{\perp }_R^{\epsilon }y$, we can obtain

$$∥x+\lambda y∥\ge ∥x∥-\epsilon ∥\lambda y∥.$$

Subsequently, we can acquire

$${\displaystyle G_{+}\left(x,y\right)=\underset{\lambda \to 0^{+}}{lim}\frac{∥x+\lambda y∥-∥x∥}{\lambda }\ge \underset{\lambda \to 0^{+}}{lim}-\frac{\epsilon ∥\lambda y∥}{\lambda }=-\epsilon }$$

and

$${\displaystyle G_{+}\left(x,y\right)=\underset{\lambda \to 0^{+}}{lim}\frac{∥x+\lambda y∥-∥x∥}{\lambda }\le \underset{\lambda \to 0^{+}}{lim}-\frac{\epsilon ∥\lambda y∥}{\lambda }=\epsilon .}$$

In conclusion, $G\left(x,y\right)\in \left[-\epsilon ,\epsilon \right]$ is evident. □

Theorem 5. 

Let X be a normed linear space, $x,y\in S_X.$ If $x{\perp }_R^{\epsilon }y$ and the norm at x are Gâteaux differentiable along the y direction, then $G\left(x,y\right)\in \left[-2\epsilon ,2\epsilon \right]$.

Proof of Theorem 5. 

From $x{\perp }_R^{\epsilon }y$, we infer that

$$|{∥x+\lambda y∥}^2-{∥x-\lambda y∥}^2|\le 4\epsilon ∥x∥∥\lambda y∥.$$

(1)

The following is divided into two cases.

Case 1: $∥x+\lambda y∥\ge ∥x-\lambda y∥$.

According to the inequality (1), we obtain

$${∥x-\lambda y∥}^2-{∥x+\lambda y∥}^2\ge -4\epsilon ∥x∥∥\lambda y∥.$$

(2)

From the inequality (2), we obtain

$${∥x-\lambda y∥}^2-{∥x∥}^2+{∥x∥}^2-{∥x+\lambda y∥}^2\ge -4\epsilon ∥x∥∥\lambda y∥,$$

which implies

$${∥x+\lambda y∥}^2-{∥x∥}^2\le 4\epsilon ∥x∥∥\lambda y∥-{∥x∥}^2+{∥x-\lambda y∥}^2.$$

Then, we can derive

$$\begin{array}{cc}\hfill ∥x∥G_{+}\left(x,y\right)& {\displaystyle =\underset{\lambda \to 0^{+}}{lim}\frac{{∥x+\lambda y∥}^2-{∥x∥}^2}{2\lambda }}\hfill \\ \hfill & {\displaystyle \le \underset{\lambda \to 0^{+}}{lim}\frac{4\epsilon ∥x∥∥\lambda y∥-{∥x∥}^2+{∥x-\lambda y∥}^2}{2\lambda }}\hfill \\ \hfill & {\displaystyle \le \underset{\lambda \to 0^{+}}{lim}\frac{4\epsilon ∥x∥∥\lambda y∥-∥\lambda y∥\left(∥x-\lambda y∥-∥x∥\right)}{2\lambda }}\hfill \\ \hfill & {\displaystyle =2\epsilon ∥x∥∥y∥-\frac{1}{2}∥y∥\left(∥x∥-∥x∥\right)=2\epsilon ,}\hfill \end{array}$$

that is, $G_{+}\left(x,y\right)\le 2\epsilon$. Similarly, we can derive

$$\begin{array}{cc}\hfill ∥x∥G_{-}\left(x,y\right)& {\displaystyle =\underset{\lambda \to 0^{-}}{lim}\frac{{∥x+\lambda y∥}^2-{∥x∥}^2}{2\lambda }}\hfill \\ \hfill & {\displaystyle \ge \underset{\lambda \to 0^{-}}{lim}\frac{4\epsilon ∥x∥∥\lambda y∥-{∥x∥}^2+{∥x-\lambda y∥}^2}{2\lambda }}\hfill \\ \hfill & {\displaystyle \ge \underset{\lambda \to 0^{-}}{lim}\frac{4\epsilon ∥x∥∥\lambda y∥+∥\lambda y∥\left(∥x-\lambda y∥-∥x∥\right)}{2\lambda }}\hfill \\ \hfill & {\displaystyle =-2\epsilon ∥x∥∥y∥+\frac{1}{2}∥y∥\left(∥x∥-∥x∥\right)=-2\epsilon ,}\hfill \end{array}$$

that is, $G_{-}\left(x,y\right)\ge -2\epsilon$.

Because the norm at x is Gâteaux differentiable along the y direction, then $G\left(x,y\right)\in \left[-2\epsilon ,2\epsilon \right]$.

Case 2: $∥x+\lambda y∥\le ∥x-\lambda y∥$.

According to the inequality (1), we obtain

$$4\epsilon ∥x∥∥\lambda y∥\ge {∥x-\lambda y∥}^2-{∥x+\lambda y∥}^2.$$

(3)

From the inequality (3), we obtain

$${∥x-\lambda y∥}^2-{∥x∥}^2+{∥x∥}^2-{∥x+\lambda y∥}^2\le 4\epsilon ∥x∥∥\lambda y∥,$$

which implies

$${∥x+\lambda y∥}^2-{∥x∥}^2\ge -4\epsilon ∥x∥∥\lambda y∥-{∥x∥}^2+{∥x-\lambda y∥}^2.$$

Then,

$$\begin{array}{cc}\hfill ∥x∥G_{+}\left(x,y\right)& {\displaystyle =\underset{\lambda \to 0^{+}}{lim}\frac{{∥x+\lambda y∥}^2-{∥x∥}^2}{2\lambda }}\hfill \\ \hfill & {\displaystyle \ge \underset{\lambda \to 0^{+}}{lim}\frac{-4\epsilon ∥x∥∥\lambda y∥-{∥x∥}^2+{∥x-\lambda y∥}^2}{2\lambda }}\hfill \\ \hfill & {\displaystyle \ge \underset{\lambda \to 0^{+}}{lim}\frac{-4\epsilon ∥x∥∥\lambda y∥+∥\lambda y∥\left(∥x-\lambda y∥-∥x∥\right)}{2\lambda }}\hfill \\ \hfill & {\displaystyle =-2\epsilon ∥x∥∥y∥+\frac{1}{2}∥y∥\left(∥x∥-∥x∥\right)=-2\epsilon ,}\hfill \end{array}$$

that is, $G_{+}\left(x,y\right)\ge -2\epsilon$. Similarly, we can obtain

$$\begin{array}{cc}\hfill ∥x∥G_{-}\left(x,y\right)& {\displaystyle =\underset{\lambda \to 0^{-}}{lim}\frac{{∥x+\lambda y∥}^2-{∥x∥}^2}{2\lambda }}\hfill \\ \hfill & {\displaystyle \le \underset{\lambda \to 0^{-}}{lim}\frac{-4\epsilon ∥x∥∥\lambda y∥-{∥x∥}^2+{∥x-\lambda y∥}^2}{2\lambda }}\hfill \\ \hfill & {\displaystyle \le \underset{\lambda \to 0^{-}}{lim}\frac{-4\epsilon ∥x∥∥\lambda y∥-∥\lambda y∥\left(∥x∥-∥x-\lambda y∥\right)}{2\lambda }}\hfill \\ \hfill & ={\displaystyle 2\epsilon ∥x∥∥y∥-\frac{1}{2}∥y∥\left(∥x∥-∥x∥\right)=2\epsilon },\hfill \end{array}$$

that is, $G_{-}\left(x,y\right)\le 2\epsilon$.

Because the norm at x is Gâteaux differentiable along the y direction, then $G\left(x,y\right)\in \left[-2\epsilon ,2\epsilon \right]$.

In summary, when $x{\perp }_R^{\epsilon }y$ and the norm at x are Gâteaux differentiable along the y direction, we can obtain $G\left(x,y\right)\in \left[-2\epsilon ,2\epsilon \right]$. □

3.2. The Angle between Vectors in a Normed Linear Space

Definition 1. 

Let X be a normed linear space, $x,y\in S_X$. The norm $∥·∥$ is Gâteaux differentiable along the direction y at x, and $G\left(x,y\right)$ exists. The cosine of the angle between x and y is the value of θ, the angle between x and y is $\theta =arccosG\left(x,y\right)$.

From Theorems 1 and 2, it follows that if x and y satisfy the relation ${\perp }_B$, ${\perp }_R$, ${\perp }_P$, ${\perp }_I$, ${\perp }_L$, ${\perp }_{\lambda P}$, then the angle between x and y is ${\displaystyle \frac{\pi }{2}}$. However, the situation is different if x and y are approximately orthogonality. According to Theorems 3 and 4, when x and y satisfy the relations ${\perp }_B^{\epsilon }$ and ${\perp }_{B\epsilon }$, the angle between x and y is $\theta \in \left[arccos\left(-\epsilon \right),arccos\epsilon \right]$.

Definition 2. 

Let X be a smooth Minkowski plane, $x,y\in X$. Take the line where the vector x is as the x-axis, the positive direction of the x-axis is the direction of the vector x, the line where the vector y is as the y-axis, and the positive direction of the y-axis is the direction of the vector y, x and y satisfy $x{\perp }_{\Delta }y$ (${\perp }_{\Delta }$ includes ${\perp }_B$, ${\perp }_R$, ${\perp }_P$, ${\perp }_I$, ${\perp }_L$,${\perp }_{\lambda P}$). Only one orthogonality can exist in this coordinate system; then, x and y form an ${\perp }_{\Delta }$ coordinate system.

Remark 1. 

For any vector z in X, there exists $\alpha ,\beta \in R$ such that $z=\alpha x+\beta y$ satisfies the relation:

$$G^2\left({\displaystyle \frac{z}{∥z∥},x}\right)+G^2\left({\displaystyle \frac{z}{∥z∥},y}\right)=1.$$

If the ${\perp }_{\Delta }$ coordinate system is the ${\perp }_I$ coordinate system, for any vectors a and b in X. If the equation $∥a+b∥=∥a-b∥$ is satisfied, then the vectors a and b are said to be orthogonal in this coordinate system and the angle between them is ${\displaystyle \frac{\pi }{2}}$, the angle θ between x and y in this coordinate system has the value range $\left[0,2\pi \right]$. When selecting a coordinate system, it is important to consider its properties and how they relate to generalized orthogonality in different situations.

Some classical orthogonality coordinate systems are selected and some examples are given below:

Example 1. 

When ${\perp }_{\Delta }$ is selected as the ${\perp }_I$ coordinate system and the norm $∥·∥$ is ${∥·∥}_1$, our orientation vector x is $\left(1,0\right)$; then, the orientation vector y is $\left(0,\pm 1\right)$. The angle between vector x and vector y can be obtained by calculating ${90}^{\circ }$.

Example 2. 

When ${\perp }_{\Delta }$ is selected as the ${\perp }_I$ coordinate system and the norm $∥·∥$ is ${∥·∥}_{\infty }$, our orientation vector x is $\left(1,0\right)$. The orientation vector y is $\left(0,\pm 1\right)$, and the angle between vector x and vector y can be calculated as ${90}^{\circ }$.

Example 3. 

When ${\perp }_{\Delta }$ is chosen as the ${\perp }_B$ coordinate system and the norm $∥·∥$ is ${∥·∥}_{\infty }$, our orientation vector x is $\left(1,0\right)$. The orientation vector y is $\left(a,\pm 1\right)$, where $a\in \left(-1,1\right)$, the calculated angle between vector x and vector y is ${90}^{\circ }$.

Example 4. 

When ${\perp }_{\Delta }$ is selected as the ${\perp }_B$ coordinate system and the norm $∥·∥$ is ${∥·∥}_1$, our orientation vector x is $\left(1,0\right)$. The orientation vector y is $\left(0,\pm 1\right)$, and the angle between vector x and vector y can be calculated as ${90}^{\circ }$.

Example 5. 

When ${\perp }_{\Delta }$ is selected as the ${\perp }_P$ coordinate system and the norm $∥·∥$ is ${∥·∥}_2$, our orientation vector x is $\left(1,0\right)$. The orientation vector y is $\left(0,\pm 1\right)$, and the angle between vector x and vector y can be calculated as ${90}^{\circ }$.

4. Conclusions

The Gâteaux derivative of the norm in a normed linear space is investigated in this paper under the framework of generalized orthogonality and approximate orthogonality theories. By applying the definition of the norm’s Gâteaux derivative, we obtain the value of the norm’s Gâteaux derivative at a point for various kinds of generalized orthogonality. Likewise, we derive the value range of the norm’s Gâteaux derivative at a point by combining different kinds of approximate orthogonality with the definition of the norm’s Gâteaux derivative. Moreover, we introduce the concept of the angle between two vectors in normed linear space and construct a ${\perp }_{\Delta }$ coordinate system on a smooth Minkowski plane. We also present several examples of classical orthogonality and ${\perp }_{\Delta }$ coordinate systems. In future work, we aim to further explore the interplay between generalized orthogonality and approximate orthogonality and the norm’s Gâteaux derivative, as well as more applications of ${\perp }_{\Delta }$ coordinate systems on the Minkowski plane, to deepen our insight into generalized positive intersection.