New Kind of orthogonality in Normed spaces by using ϼ<i>*</i> -orthogonality

Laith Ahmed Jasim - ليث احمد جاسم محمد; Mohammed Yahya Abed

doi:10.12688/f1000research.174190.1

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Research Article

New Kind of orthogonality in Normed spaces by using ϼ* -orthogonality

[version 1; peer review: awaiting peer review]

Laith Ahmed Jasim - ليث احمد جاسم محمد ¹, Mohammed Yahya Abed²

PUBLISHED 31 Dec 2025

Author details Author details

¹ University of Kerbala, Karbala, Karbala Governorate, Iraq
² University of Kerbala, Karbala, Karbala Governorate, Iraq

Laith Ahmed Jasim - ليث احمد جاسم محمد
Roles: Project Administration, Writing – Original Draft Preparation, Writing – Review & Editing

Mohammed Yahya Abed
Roles: Conceptualization, Methodology, Project Administration, Supervision, Validation, Writing – Review & Editing

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REVIEWER STATUS AWAITING PEER REVIEW

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

Orthogonality is an important concept in normed spaces, and many generalizations of this concept have been proposed. In this paper, we introduce a new concept of orthogonality in normed spaces, denoted by ϼ LM -orthogonality, which is related to ϼ ∗ -orthogonality. The fundamental properties of this new concept are systematically analyzed, including symmetry, non-negativity, scalar homogeneity, linear independence of orthogonal vectors, the Cauchy–Schwarz type inequality, the zero property, and other characteristics. We also explain the relationship between this new definition and some other well-known types of orthogonality.

Keywords

orthogonality, norm derivatives, ϼ_*-orthogonality.

Corresponding author: Laith Ahmed Jasim - ليث احمد جاسم محمد

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2025 Ahmed Jasim - ليث احمد جاسم محمد L and Yahya Abed M. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Ahmed Jasim - ليث احمد جاسم محمد L and Yahya Abed M. New Kind of orthogonality in Normed spaces by using ϼ* -orthogonality [version 1; peer review: awaiting peer review]. F1000Research 2025, 14:1486 (https://doi.org/10.12688/f1000research.174190.1) First published: 31 Dec 2025, 14:1486 (https://doi.org/10.12688/f1000research.174190.1) Latest published: 31 Dec 2025, 14:1486 (https://doi.org/10.12688/f1000research.174190.1)

1. Introduction

In an inner product space $(\dot{X}, ⟨ \cdot, \cdot ⟩)$ , two vectors $\dot{s}, \dot{r} in \dot{X}$ are said to be orthogonal, denoted by $\dot{s} ⊥ \dot{r}$ if and only if $⟨ \dot{s}, \dot{r} ⟩ = 0$ . In spaces whose norm is induced by an inner product, the notion of orthogonality admits a natural and uniquely determined form. Extending this concept to more general normed spaces enable a deeper investigation of their geometric and analytical interrelations, thus offering a broader framework to generalize classical Euclidean ideas to abstract mathematical settings.

Consider $(\dot{X}, ‖ . ‖)$ as a real normed vector space whose dimension is not less than two, there exist several notions of orthogonality, among which the definition introduced by (Birkhoff–James) is one of the most prominent. If $\dot{s}, \dot{r} \in \dot{X}$ , define $\dot{s} ⊥_{B} \dot{r}$ ⇔ $‖ \dot{s} + t \dot{r} ‖ \geq ‖ \dot{s} ‖$ , (t $\in R$ ) see.¹

In 1986, Amir in² defined the norm derivatives using two functionals, $ϼ_{-}, ϼ_{+} : \dot{X} \times \dot{X} \to ℝ$ be functionals, as follows:

ϼ_{-} (\dot{s}, \dot{r}) = ‖ \dot{s} ‖ lim_{t \to 0^{-}} \frac{‖ \dot{s} + t \dot{r} ‖ - ‖ \dot{s} ‖}{t} and ϼ_{+} (\dot{s}, \dot{r}) = ‖ \dot{s} ‖ lim_{t \to 0^{+}} \frac{‖ \dot{s} + t \dot{r} ‖ - ‖ \dot{s} ‖}{t}

then ṡ is

ϼ_{-}

- orthogonality to

\dot{r}

denoted by

\dot{s} ⊥_{ϼ_{-}} \dot{r} ⟺

ϼ_{-} (\dot{s}, \dot{r}) = 0

and ṡ is

ϼ_{+}

- orthogonality to

\dot{r}

denoted by

\dot{s} ⊥_{ϼ_{+}} \dot{r} ⟺

ϼ_{+} (\dot{s}, \dot{r}) = 0

.

A mapping ${⟨ \cdot, \cdot ⟩}_{g} : \dot{X} \times \dot{X} \to ℝ$ was introduced by MiliĆiĆ in³ as follows: $ϼ (\dot{s}, \dot{r}) = {⟨ \dot{s}, \dot{r} ⟩}_{g} = \frac{ϼ_{-} (\dot{s}, \dot{r}) + ϼ_{+} (\dot{s}, \dot{r})}{2}$ . But M. Nur and H. Gunawan in⁴ defined an orthogonality relation based on norm derivatives by $ϼ_{gg} = \sqrt{| ϼ (\dot{s}, \dot{r}) | | ϼ (\dot{r}, \dot{s}) |}$ , Also Zamani and Moslehian in⁵ defined an orthogonality functional $ϼ_{λ} : : \dot{X} \times \dot{X} \to ℝ$ by $ϼ_{λ}$ (ṡ, $\dot{r}$ ) = λ $ϼ_{-} (\dot{s}, \dot{r})$ + (1- λ) $ϼ_{+} (\dot{s}, \dot{r})$ ∀ λ $\in$ R, and ṡ is $ϼ_{λ}$ - orthogonality to $\dot{r}$ denoted by $\dot{s} ⊥_{ϼ_{λ}} \dot{r} ⟺$ $ϼ_{λ} (\dot{s}, \dot{r}) = 0$ the notions of $ϼ_{*}$ -orthogonality is introduced in⁶ as: $\dot{s} ⊥_{ϼ_{*}} \dot{r} ⟺$ $ϼ_{*} (\dot{s}, \dot{r}) = ϼ_{+} (\dot{s}, \dot{r}) ϼ_(\dot{s}, \dot{r}) =$ 0. Recall that $\dot{X}$ is smooth at $\dot{s}$ if and only if $ϼ_(\dot{s}, \dot{r}) = ϼ_{+} (\dot{s}, \dot{r})$ for all $\dot{r} \in \dot{X} .$

These functionals extend the concept of the inner product to a normed space, and many geometric properties will be reformulated in the normed space using the concept of the norm derivative. Now we shall address some important theorem that we will need in this research.

Theorem 1.1:

⁶Suppose ( $\dot{X}$ , ‖·‖) represents a real normed vector space, then

(i) $ϼ_{*}$ (t ṡ, $\dot{r}$ ) = $ϼ_{*}$ (ṡ, t $\dot{r}$ )= $t^{2} ϼ_{*}$ (ṡ, $\dot{r}$ ) for all ṡ, $\dot{r}$ ∈ $\dot{X}$ and all t ∈ R.
(ii) | $ϼ_{*} (\dot{s}, \dot{r})$ | $\leq {‖ \dot{s} ‖}^{2} {‖ \dot{r} ‖}^{2}$ for all ṡ, $\dot{r} \in \dot{X}$ .
(iii) For all vectors except the zero vector, ṡ, $\dot{r}$ ∈ $\dot{X}$ , if $\dot{s}$ $⊥_{ϼ_{*}}$ $\dot{r}$ , then ṡ and $\dot{r}$ are independent in the linear sense.
(iv) $ϼ_{*}$ (ṡ, t ṡ + $\dot{r}$ ) = $t^{2} {‖ \dot{s} ‖}^{4}$ +2t ${‖ \dot{s} ‖}^{2} ϼ ($ ṡ, $\dot{r}$ ) + $ϼ_{*}$ (ṡ, $\dot{r}$ ) for all ṡ, $\dot{r} \in \dot{X}$ and all t ∈ R.

For more information about other orthogonalities you can see.^7–10

2- Fundamental results

This section is devoted to introducing a new form of orthogonality by using $ϼ_{*}$ -orthogonality as follows:

Definition 2.1:

Suppose ( $\dot{X}$ , ‖·‖) represents a real normed vector space then $\forall \dot{s}, \dot{r} \in \dot{X}$ , the functional $ϼ_{LM} : \dot{X} \times \dot{X} \to R$ which defines as $ϼ_{LM} (\dot{s}, \dot{r}) = \sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |}$ , We say that ṡ is $ϼ_{LM}$ - orthogonality to $\dot{r}$ , we denoted by ṡ $⊥_{ϼ_{LM}}$ $\dot{r}$ if $ϼ_{LM} (\dot{s}, \dot{r}) = 0$ .

Proposition 2.2:

Suppose ( $\dot{X}$ , ‖·‖) represents a real normed vector space. Then

(a) $ϼ_{LM} (\dot{s}, \dot{r}) \geq 0$ for all ṡ, $\dot{r} \in \dot{X}$ . (non-negative)
(b) $ϼ_{LM} (\dot{s}, 0) = ϼ_{LM} (0, \dot{r}) = 0$ for all ṡ, $\dot{r} \in \dot{X} .$ (zero property)
(c) $ϼ_{LM} (\dot{s}, \dot{s}) = {‖ \dot{s} ‖}^{4}$ for all ṡ $\in \dot{X}$ , also $ϼ_{LM} (\dot{s}, \dot{s})$ = $ϼ_{*} (\dot{s}, \dot{s})$
(d) $ϼ_{LM} (\dot{s}, \dot{r}) = ϼ_{LM} (\dot{r}, \dot{s})$ for all ṡ, $\dot{r} \in \dot{X} .$ (symmetry)
(e) $ϼ_{LM} (α \dot{s}, β \dot{r}) = {| α β |}^{2} ϼ_{LM} (\dot{s}, \dot{r})$ for all $\dot{s}, \dot{r} \in \dot{X}$ and all $α, β$ $\in R .$ (scalar homogeneity)
(f ) $ϼ_{LM} (\dot{s}, \dot{r}) \leq {‖ \dot{s} ‖}^{2} {‖ \dot{r} ‖}^{2}$ for all ṡ, $\dot{r} \in \dot{X} .$ (Cauchy schwarz inequality)
(g) $ϼ_{LM} (\dot{s}, t \dot{s} + \dot{r}) =$ $\sqrt{| (t^{2} {‖ \dot{s} ‖}^{4} + 2 t {‖ \dot{s} ‖}^{2} ϼ (\dot{s}, \dot{r}) + ϼ_{*} (\dot{s}, \dot{r})) (ϼ_{*} (t \dot{s} + \dot{r}, \dot{s})) |}$
(h) For all nonzero vectors ṡ, $\dot{r} \in \dot{X}$ , if $\dot{s} ⊥_{ϼ_{LM}} \dot{r}$ , then ṡ and $\dot{r}$ are linearly independent. (linear independence)

Proof:

(a) We have $ϼ_{LM} (\dot{s}, \dot{r})$ = $\sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |}$ since $| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) | \geq$ 0 then $\sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |}$ $\geq$ 0, implies $ϼ_{LM} (\dot{s}, \dot{r}) \geq$ 0.
(b) From definition of $ϼ_{LM}$ -orthogonality we have $ϼ_{LM} (\dot{s}, 0)$ = $\sqrt{| ϼ_{*} (\dot{s}, 0) ϼ_{*} (0, \dot{s}) |}$ = $\sqrt{| 0.0 |}$ = 0
In the same way, we prove $ϼ_{LM} (0, \dot{r}) = 0$ .
(c) From definition of $ϼ_{*}$ -orthogonality and $ϼ_{LM}$ -orthogonality we get $ϼ_{LM} (\dot{s}, \dot{s})$ = $\sqrt{| ϼ_{*} (\dot{s}, \dot{s}) ϼ_{*} (\dot{s}, \dot{s}) |}$

= \sqrt{| ϼ_{-} (\dot{s}, \dot{s}) ϼ_{+} (\dot{s}, \dot{s}) ϼ_{-} (\dot{s}, \dot{s}) ϼ_{+} (\dot{s}, \dot{s}) |} = \sqrt{| {‖ \dot{s} ‖}^{2} {‖ \dot{s} ‖}^{2} {‖ \dot{s} ‖}^{2} {‖ \dot{s} ‖}^{2} |} = {‖ \dot{s} ‖}^{4}

(d) $ϼ_{LM} (\dot{s}, \dot{r})$ = $\sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |}$ = $\sqrt{| ϼ_{*} (\dot{r}, \dot{s}) ϼ_{*} (\dot{s}, \dot{r}) |}$ = $ϼ_{LM} (\dot{r}, \dot{s})$
(e) By theorem 1.1 (i) we have $ϼ_{*} (t \dot{s}, \dot{r})$ = $ϼ_{*} (\dot{s}, t \dot{r})$ = $t^{2} ϼ_{*} (\dot{s}, \dot{r})$ for all $\dot{s}, \dot{r} \in \dot{X}$ and all $t \in R$ .
$ϼ_{*} (α \dot{s}, β \dot{r})$ = α²β² $ϼ_{*} (\dot{s}, \dot{r})$ and $ϼ_{*} (β \dot{r}, α \dot{s})$ = α²β² $ϼ_{*} (\dot{r}, \dot{s})$ then
$\begin{matrix} ϼ_{LM} (α \dot{s}, β \dot{r}) = \sqrt{| ϼ_{*} (α \dot{s}, β \dot{r}) ϼ_{*} (β \dot{r}, α \dot{s}) |} = \sqrt{| α^{2} β^{2} ϼ_{*} (\dot{s}, \dot{r}) α^{2} β^{2} ϼ_{*} (\dot{r}, \dot{s}) |} \\ = {| α β |}^{2} \sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |} = {| α β |}^{2} ϼ_{LM} (\dot{s}, \dot{r}) \end{matrix}$
(f ) By theorem 1.1 (ii) we have | $ϼ_{*} (\dot{s}, \dot{r})$ )| $\leq {‖ \dot{s} ‖}^{2} {‖ \dot{r} ‖}^{2}$ for all $\dot{s}, \dot{r} \in \dot{X}$
Then $ϼ_{LM} (\dot{s}, \dot{r}) = \sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |}$ $\leq \sqrt{{‖ \dot{s} ‖}^{2} {‖ \dot{r} ‖}^{2} {‖ \dot{r} ‖}^{2} {‖ \dot{s} ‖}^{2}}$ = ${‖ \dot{s} ‖}^{2} {‖ \dot{r} ‖}^{2}$ for all $\dot{s}, \dot{r} \in \dot{X}$
Hence $ϼ_{LM} (\dot{s}, \dot{r}) \leq {‖ \dot{s} ‖}^{2} {‖ \dot{r} ‖}^{2}$ for all ṡ, $\dot{r} \in \dot{X}$
(g) By theorem 1.1 (iv) we have $ϼ_{*} (\dot{s}, t \dot{s} + \dot{r})$ = $t^{2} {‖ \dot{s} ‖}^{4}$ +2t ${‖ \dot{s} ‖}^{2} ϼ (\dot{s}, \dot{r})$ + $ϼ_{*} (\dot{s}, \dot{r})$
$ϼ_{LM} (\dot{s}, t \dot{s} + \dot{r}) = \sqrt{| ϼ_{*} (\dot{s}, t \dot{s} + \dot{r}) ϼ_{*} (t \dot{s} + \dot{r}, \dot{s}) |} = \sqrt{| (t^{2} {‖ \dot{s} ‖}^{4} + 2 t {‖ \dot{s} ‖}^{2} ϼ (\dot{s}, \dot{r}) + ϼ_{*} (\dot{s}, \dot{r})) (ϼ_{*} (t \dot{s} + \dot{r}, \dot{s})) |}$
(h) we have ṡ $⊥_{ϼ_{LM}}$ $\dot{r}$ therefore; $ϼ_{LM} (\dot{s}, \dot{r}) = \sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |}$ = 0

Thus $ϼ_{*} (\dot{s}, \dot{r})$ = 0 or $ϼ_{*} (\dot{r}, \dot{s})$ = 0

By theorem 1.1 (iii) for all nonzero vectors ṡ, $\dot{r} \in \dot{X}$ , if $\dot{s}$ $⊥_{ϼ_{*}}$ $\dot{r}$ , then ṡ and $\dot{r}$ are linearly independent.

Hence, whether $ϼ_{*} (\dot{s}, \dot{r})$ = 0 or $ϼ_{*} (\dot{r}, \dot{s})$ = 0 it follows that ṡ and $\dot{r}$ are linearly independent.

Remark 2.3:

If $\dot{X}$ come from inner product space then $ϼ_{LM} (\dot{s}, \dot{r})$ = $ϼ_{*} (\dot{s}, \dot{r})$ .

3. The connection between $ϼ_{LM}$ – orthogonality and some others orthogonalities

Theorem 3.1:

Suppose $(\dot{X}, ‖ \cdot ‖)$ represents a real normed vector space and let $\dot{s}$ , $\dot{r}$ ∈ $\dot{X}$ .

(a) If $\dot{s} ⊥_{ϼ_{*}} \dot{r}$ , then $\dot{s} ⊥_{ϼ_{LM}} \dot{r}$
(b) If $\dot{s} ⊥_{ϼ_{-}} \dot{r}$ , then $\dot{s} ⊥_{ϼ_{LM}} \dot{r}$
(c) If $\dot{s} ⊥_{ϼ_{+}} \dot{r}$ , then $\dot{s} ⊥_{ϼ_{LM}} \dot{r}$
(d) If $\dot{r} ⊥_{ϼ_{*}} \dot{s}$ , then $\dot{s} ⊥_{ϼ_{LM}} \dot{r}$
(e) If $\dot{r} ⊥_{ϼ_{-}} \dot{s}$ , then $\dot{s} ⊥_{ϼ_{LM}} \dot{r}$
(f ) If $\dot{r} ⊥_{ϼ_{+}} \dot{s}$ , then $\dot{s} ⊥_{ϼ_{LM}} \dot{r}$

Proof:

Obvious

Remark 3.2:

The converse of (a), (b) and (c) of theorem 3.1 are not true. For example, take $\dot{s} = (0, 2)$ and $\dot{r} = (1, 1)$ if $‖ ({\dot{s}}_{1}, {\dot{s}}_{2}) ‖ = max {| {\dot{s}}_{1} |, | {\dot{s}}_{2} |}$ in the space $\dot{X}$ = $R^{2}$

\begin{matrix} ϼ_{+} (\dot{s}, \dot{r}) = ‖ \dot{s} ‖ lim_{t \to 0^{+}} \frac{‖ \dot{s} + t \dot{r} ‖ - ‖ \dot{s} ‖}{t} = (2) lim_{t \to 0^{+}} \frac{| 2 + t | - 2}{t} = 2 (1) = 2 \neq 0 \\ ϼ_{-} (\dot{s}, \dot{r}) = ‖ \dot{s} ‖ lim_{t \to 0^{-}} \frac{‖ \dot{s} + t \dot{r} ‖ - ‖ \dot{s} ‖}{t} = (2) lim_{t \to 0^{-}} \frac{| 2 + t | - 2}{t} = 2 (1) = 2 \neq 0 \\ \begin{matrix} ϼ_{+} (\dot{r}, \dot{s}) = ‖ \dot{r} ‖ lim_{t \to 0^{+}} \frac{‖ \dot{r} + t \dot{s} ‖ - ‖ \dot{r} ‖}{t} = (1) lim_{t \to 0^{+}} \frac{| 1 + 2 t | - 1}{t} = 1 (2) = 2 \\ ϼ_{-} (\dot{r}, \dot{s}) = ‖ \dot{r} ‖ lim_{t \to 0^{-}} \frac{‖ \dot{r} + t \dot{s} ‖ - ‖ \dot{r} ‖}{t} = (1) lim_{t \to 0^{-}} \frac{| 1 | - 1}{t} = \end{matrix} \end{matrix}

Since $ϼ_{*} (\dot{s}, \dot{r}) = ϼ_{+} (\dot{s}, \dot{r}) ϼ_(\dot{s}, \dot{r})$

ϼ_{*} (\dot{s}, \dot{r}) = (2) (2) = 4 \neq 0

But $ϼ_{LM} (\dot{s}, \dot{r}) = \sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |}$ = 0

Hence ṡ $⊥_{ϼ_{LM}} \dot{r}$ but the others are not.

Remark 3.3:

The converse of (d), (e) and (f ) of theorem 3.1 are not true. For example, take ṡ = (2,0,0) and $\dot{r}$ = (1,1,0) if $‖ ({\dot{s}}_{1}, {\dot{s}}_{2}, {\dot{s}}_{3}) ‖$ = $| {\dot{s}}_{1} | + | {\dot{s}}_{2} |$ + $| {\dot{s}}_{3} |$ in the space X = $R^{3}$ , then

\begin{matrix} ϼ_{+} (\dot{s}, \dot{r}) = ‖ \dot{s} ‖ lim_{t \to 0^{+}} \frac{‖ \dot{s} + t \dot{r} ‖ - ‖ \dot{s} ‖}{t} = (2) lim_{t \to 0^{+}} \frac{| 2 + t | + | t | - 2}{t} = 2 (2) = 4 \\ ϼ_{-} (\dot{s}, \dot{r}) = ‖ \dot{s} ‖ lim_{t \to 0^{-}} \frac{‖ \dot{s} + t \dot{r} ‖ - ‖ \dot{s} ‖}{t} = (2) lim_{t \to 0^{-}} \frac{| 2 + t | + | t | - 2}{t} = 2 (0) = 0 \\ \begin{matrix} ϼ_{+} (\dot{r}, \dot{s}) = ‖ \dot{r} ‖ lim_{t \to 0^{+}} \frac{‖ \dot{r} + t \dot{s} ‖ - ‖ \dot{r} ‖}{t} = (2) lim_{t \to 0^{+}} \frac{| 1 + 2 t | + | 1 | - 2}{t} = 2 (2) = 4 \neq 0 \\ ϼ_{-} (\dot{r}, \dot{s}) = ‖ \dot{r} ‖ lim_{t \to 0^{-}} \frac{‖ \dot{r} + t \dot{s} ‖ - ‖ \dot{r} ‖}{t} = (2) lim_{t \to 0^{-}} \frac{| 1 + 2 t | + | 1 | - 2}{t} = 2 (2) = 4 \neq 0 \end{matrix} \end{matrix}

Since $ϼ_{*} (\dot{r}, \dot{s}) = ϼ_{+} (\dot{r}, \dot{s}) ϼ_{-} (\dot{r}, \dot{s})$

ϼ_{*} (\dot{r}, \dot{s}) = (4) (4) = 16 \neq 0

But $ϼ_{LM} (\dot{s}, \dot{r}) = \sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |}$ = 0

Hence ṡ $⊥_{ϼ_{LM}} \dot{r}$ but the others are not.

Example 3.4:

Consider the real normed space $\dot{X}$ = $R^{2}$ with the norm $‖ \dot{s} ‖$ = $| {\dot{s}}_{1} | + | {\dot{s}}_{2} |$ where $\dot{s} = (1, 0)$ ,

$\dot{r} = (0, 1)$ we have $ϼ_{+} (\dot{s}, \dot{r})$ = $‖ \dot{s} ‖ lim_{t \to 0^{+}} \frac{‖ \dot{s} + t \dot{r} ‖ - ‖ \dot{s} ‖}{t}$ and $ϼ_{-} (\dot{s}, \dot{r})$ = $‖ \dot{s} ‖ lim_{t \to 0^{-}} \frac{‖ \dot{s} + t \dot{r} ‖ - ‖ \dot{s} ‖}{t}$

Thus $ϼ_{+} (\dot{s}, \dot{r})$ = (1) $lim_{t \to 0^{+}} \frac{1 + t - 1}{t}$ = 1

$ϼ_{-} (\dot{s}, \dot{r})$ = $(1) lim_{t \to 0^{-}} \frac{1 - t - 1}{t}$ = -1

$ϼ_{+} (\dot{r}, \dot{s})$ = (1) $lim_{t \to 0^{+}} \frac{t + 1 - 1}{t}$ = 1

$ϼ_{-} (\dot{r}, \dot{s})$ = $(1) lim_{t \to 0^{-}} \frac{- t + 1 - 1}{t}$ = -1

We have $ϼ (\dot{s}, \dot{r})$ = $\frac{ϼ_{-} (\dot{s}, \dot{r}) + ϼ_{+} (\dot{s}, \dot{r})}{2}$ therefore $ϼ (\dot{s}, \dot{r})$ = $\frac{- 1 + 1}{2}$ = 0,

Since $ϼ_{gg} = \sqrt{| ϼ (\dot{s}, \dot{r}) | | ϼ (\dot{r}, \dot{s}) |}$ implies $ϼ_{gg} (\dot{s}, \dot{r})$ = $\sqrt{0}$ = 0,

Also $ϼ_{λ} (\dot{s}, \dot{r})$ = λ $ϼ_{-} (\dot{s}, \dot{r})$ + (1- λ) $ϼ_{+} (\dot{s}, \dot{r})$ ) = λ $(- 1)$ + (1- λ) (1)= 1-2λ put λ = $\frac{1}{2}$ then $ϼ_{λ} (\dot{s}, \dot{r}) = 0,$

But $ϼ_{LM} (\dot{s}, \dot{r})$ = $\sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |}$ = $\sqrt{| 1 |}$ = 1 $\neq$ 0

Hence $⊥_{ϼ}$ $⊈$ $⊥_{ϼ_{LM}}$ , $⊥_{ϼ_{gg}} ⊈ ⊥_{ϼ_{LM}}$ and $⊥_{ϼ_{λ}} ⊈$ $⊥_{ϼ_{LM}} .$

Example 3.5:

Consider the real normed space $\dot{X}$ = $R^{2}$ with the norm $‖ (\dot{s}, \dot{r}) ‖$ = max { $| \dot{s} |, | \dot{r} |$ } where $\dot{s} = (1, 1)$ , $\dot{r} = (0, - 1)$ . Then for every $\dot{s}, \dot{r} \in \dot{X}$ we have

\begin{matrix} ϼ_{+} (\dot{s}, \dot{r}) = (1) lim_{t \to 0^{+}} \frac{| 1 | - 1}{t} = 1 (0) = 0 \\ ϼ_{-} (\dot{s}, \dot{r}) = (1) lim_{t \to 0^{-}} \frac{| 1 - t | - 1}{t} = 1 (- 1) = - 1 \\ \begin{matrix} ϼ_{+} (\dot{r}, \dot{s}) = (1) lim_{t \to 0^{+}} \frac{| t | - 1}{t} = 1 (0) = 0 \\ ϼ_{-} (\dot{r}, \dot{s}) = (1) lim_{t \to 0^{-}} \frac{| - 1 + t | - 1}{t} = lim_{t \to 0^{-}} \frac{| t - 1 | - 1}{t} = - 1 \end{matrix} \end{matrix}

We have $ϼ (\dot{s}, \dot{r})$ = $\frac{ϼ_{-} (\dot{s}, \dot{r}) + ϼ_{+} (\dot{s}, \dot{r})}{2}$ therefore $ϼ (\dot{s}, \dot{r})$ = $\frac{- 1 + 0}{2}$ = $\frac{- 1}{2}$ $\neq$ 0 and $ϼ (\dot{r}, \dot{s})$ = $\frac{ϼ_{-} (\dot{r}, \dot{s}) + ϼ_{+} (\dot{r}, \dot{s})}{2}$ = $\frac{- 1 + 0}{2}$ = $\frac{- 1}{2}$

ϼ_{gg} (\dot{s}, \dot{r}) = \sqrt{(\frac{- 1}{2}) (\frac{- 1}{2})} = \sqrt{(\frac{1}{4})} = \frac{1}{2} \neq 0

ϼ_{λ} (ṡ, \dot{r}) = λ ϼ_{-} (\dot{s}, \dot{r}) + (1 - λ) ϼ_{+} (\dot{s}, \dot{r})) = λ (- 1) + (1 - λ) (0) = - λ \neq 0

but

ϼ_{LM}

(ṡ,

\dot{r}

) =

\sqrt{| ϼ_{*} (\dot{s}, \dot{r}) ϼ_{*} (\dot{r}, \dot{s}) |}

= 0.

Hence $⊥_{ϼ_{LM}}$ $⊈$ $⊥_{ϼ}$ , $⊥_{ϼ_{LM}} ⊈ ⊥_{ϼ_{gg}}$ and $⊥_{ϼ_{LM}} ⊈$ $⊥_{ϼ_{λ}}$

Remark 3.6:

From example 3.4 and example 3.5 it follows that the orthogonalities, $⊥_{ϼ}$ , $⊥_{ϼ_{gg}}$ , $⊥_{ϼ_{λ}}$ and $⊥_{ϼ_{LM}}$ are generally incomparable.

Remark 3.7:

If X is smooth then $⊥_{ϼ_{-}} = ⊥_{ϼ_{+}} = ⊥_{ϼ} = ⊥_{ϼ_{*}} = ⊥_{ϼ_{λ}}$ but not equal to $⊥_{ϼ_{LM}}$ .

Ethical considerations

This study is entirely theoretical and does not involve any human participants or animals, nor does it rely on data requiring ethical approval. Therefore, no ethical concerns arise in conducting this research.

Data availability

This study is entirely theoretical and does not rely on any experimental or external datasets. Therefore, no data are available in connection with this research.

Reporting guidelines

This article is a purely theoretical study in functional analysis and does not involve clinical trials, animal studies, observational studies, or qualitative research. Therefore, standard reporting guidelines such as CONSORT, ARRIVE, STROBE, or COREQ/SRQR do not directly apply. Standard practices for presenting theoretical mathematical results have been followed, including clear definitions, theorems, proofs, and logical consistency.

References

1. Birkhoff G: Orthogonality in linear metric spaces. Duke Mathematical Journal. 1935; 1(2): 169–172. Publisher Full Text
2. Amir D: Characterizations of Inner Products Spaces. Basel: Birkauser Verlag; 1986.
3. Miličić PM: Sur la G-orthogonalité dans les espaces normés. Mat. Vesn. 1987; 39: 325–334.
4. Nur M, Gunawan H: A new orthogonality and angle in a normed space. Aequationes Math. 2018; 93(1): 1–18.
5. Moslehian MS, Zamani A: An extension of orthogonality relations based on norm derivatives. Q. J. Math. 2018; 70(2): 379–393.
6. Moslehian MS, Zamani A, Dehghani M: Characterizations of smooth spaces by ϼ∗-orthogonality. Houst. J. Math. 2017; 43(4): 1061–1074.
7. Chmieliński J, Wójcik P: On a ϼ -orthogonality. Aequationes Math. 2010; 80(1): 45–55. Publisher Full Text
8. Abed MY: New form of orthogonality in the real normed spaces. Journal of Interdisciplinary Mathematics. 2022; 25(8): 2579–2584. Publisher Full Text
9. Abed MY: On semi and quasi γϼϼ-orthogonality in real normed space. Journal of Interdisciplinary Mathematics. 2022b; 25(8): 2585–2590. Publisher Full Text
10. Alonso J, Benítez C: Orthogonality in normed linear spaces: A survey. Part II: Relations between main orthogonalities. Extracta Mathematicae. 1989; 4(3): 121–131.

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¹ University of Kerbala, Karbala, Karbala Governorate, Iraq
² University of Kerbala, Karbala, Karbala Governorate, Iraq

Laith Ahmed Jasim - ليث احمد جاسم محمد
Roles: Project Administration, Writing – Original Draft Preparation, Writing – Review & Editing

Mohammed Yahya Abed
Roles: Conceptualization, Methodology, Project Administration, Supervision, Validation, Writing – Review & Editing

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No competing interests were disclosed.

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The author(s) declared that no grants were involved in supporting this work.

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Ahmed Jasim - ليث احمد جاسم محمد L and Yahya Abed M. New Kind of orthogonality in Normed spaces by using ϼ* -orthogonality [version 1; peer review: awaiting peer review]. F1000Research 2025, 14:1486 (https://doi.org/10.12688/f1000research.174190.1)

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[1] 1. Birkhoff G: Orthogonality in linear metric spaces. Duke Mathematical Journal. 1935; 1(2): 169–172. Publisher Full Text

[2] 2. Amir D: Characterizations of Inner Products Spaces. Basel: Birkauser Verlag; 1986.

[3] 3. Miličić PM: Sur la G-orthogonalité dans les espaces normés. Mat. Vesn. 1987; 39: 325–334.

[4] 4. Nur M, Gunawan H: A new orthogonality and angle in a normed space. Aequationes Math. 2018; 93(1): 1–18.

[5] 5. Moslehian MS, Zamani A: An extension of orthogonality relations based on norm derivatives. Q. J. Math. 2018; 70(2): 379–393.

[6] 6. Moslehian MS, Zamani A, Dehghani M: Characterizations of smooth spaces by ϼ∗-orthogonality. Houst. J. Math. 2017; 43(4): 1061–1074.

[7] 7. Chmieliński J, Wójcik P: On a ϼ -orthogonality. Aequationes Math. 2010; 80(1): 45–55. Publisher Full Text

[8] 8. Abed MY: New form of orthogonality in the real normed spaces. Journal of Interdisciplinary Mathematics. 2022; 25(8): 2579–2584. Publisher Full Text

[9] 9. Abed MY: On semi and quasi γϼϼ-orthogonality in real normed space. Journal of Interdisciplinary Mathematics. 2022b; 25(8): 2585–2590. Publisher Full Text

[10] 10. Alonso J, Benítez C: Orthogonality in normed linear spaces: A survey. Part II: Relations between main orthogonalities. Extracta Mathematicae. 1989; 4(3): 121–131.

New Kind of orthogonality in Normed spaces by using ϼ* -orthogonality

Abstract

Keywords

1. Introduction

Theorem 1.1:

2- Fundamental results

Definition 2.1:

Proposition 2.2:

Proof:

Remark 2.3:

3. The connection between ϼLM – orthogonality and some others orthogonalities

Theorem 3.1:

Proof:

Remark 3.2:

Remark 3.3:

Example 3.4:

Example 3.5:

Remark 3.6:

Remark 3.7:

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3. The connection between $ϼ_{LM}$ – orthogonality and some others orthogonalities