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

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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." } { "@context": "http://schema.org", "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": "1", "item": { "@id": "https://f1000research.com/", "name": "Home" } }, { "@type": "ListItem", "position": "2", "item": { "@id": "https://f1000research.com/browse/articles", "name": "Browse" } }, { "@type": "ListItem", "position": "3", "item": { "@id": "https://f1000research.com/articles/14-1486/v1", "name": "New Kind of orthogonality in Normed spaces by using ϼ* -orthogonality" } } ] } Home Browse New Kind of orthogonality in Normed spaces by using ϼ* -orthogonality ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article 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: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.12688/f1000research.174190.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article New Kind of orthogonality in Normed spaces by using ϼ * -orthogonality [version 1; peer review: 2 approved, 1 approved with reservations] Laith Ahmed Jasim - ليث احمد جاسم محمد https://orcid.org/0009-0000-6299-7094 1 , Mohammed Yahya Abed 2 Laith Ahmed Jasim - ليث احمد جاسم محمد https://orcid.org/0009-0000-6299-7094 1 , Mohammed Yahya Abed 2 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 OPEN PEER REVIEW DETAILS REVIEWER STATUS 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. READ ALL READ LESS Keywords orthogonality, norm derivatives, ϼ_*-orthogonality. Corresponding Author(s) Laith Ahmed Jasim - ليث احمد جاسم محمد ( [email protected] ) Close 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: 2 approved, 1 approved with reservations] . 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 ( X ̇ , ⟨ ⋅ , ⋅ ⟩ ) , two vectors s ̇ , r ̇ in X ̇ are said to be orthogonal, denoted by s ̇ ⊥ r ̇ if and only if ⟨ s ̇ , 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 ( 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 s ̇ , r ̇ ∈ X ̇ , define s ̇ ⊥ B r ̇ ⇔ ‖ s ̇ + t r ̇ ‖ ≥ ‖ s ̇ ‖ , (t ∈ R ) see. 1 In 1986, Amir in 2 defined the norm derivatives using two functionals, ϼ − , ϼ + : X ̇ × X ̇ → ℝ be functionals, as follows: ϼ − ( s ̇ , r ̇ ) = ‖ s ̇ ‖ lim t → 0 − ‖ s ̇ + t r ̇ ‖ − ‖ s ̇ ‖ t and ϼ + ( s ̇ , r ̇ ) = ‖ s ̇ ‖ lim t → 0 + ‖ s ̇ + t r ̇ ‖ − ‖ s ̇ ‖ t then ṡ is ϼ − - orthogonality to r ̇ denoted by s ̇ ⊥ ϼ − r ̇ ⟺ ϼ − ( s ̇ , r ̇ ) = 0 and ṡ is ϼ + - orthogonality to r ̇ denoted by s ̇ ⊥ ϼ + r ̇ ⟺ ϼ + ( s ̇ , r ̇ ) = 0 . A mapping ⟨ ⋅ , ⋅ ⟩ g : X ̇ × X ̇ → ℝ was introduced by MiliĆiĆ in 3 as follows: ϼ ( s ̇ , r ̇ ) = ⟨ s ̇ , r ̇ ⟩ g = ϼ − ( s ̇ , r ̇ ) + ϼ + ( s ̇ , r ̇ ) 2 . But M. Nur and H. Gunawan in 4 defined an orthogonality relation based on norm derivatives by ϼ gg = | ϼ ( s ̇ , r ̇ ) | | ϼ ( r ̇ , s ̇ ) | , Also Zamani and Moslehian in 5 defined an orthogonality functional ϼ λ : : X ̇ × X ̇ → ℝ by ϼ λ (ṡ, r ̇ ) = λ ϼ − ( s ̇ , r ̇ ) + (1- λ) ϼ + ( s ̇ , r ̇ ) ∀ λ ∈ R, and ṡ is ϼ λ - orthogonality to r ̇ denoted by s ̇ ⊥ ϼ λ r ̇ ⟺ ϼ λ ( s ̇ , r ̇ ) = 0 the notions of ϼ ∗ -orthogonality is introduced in 6 as: s ̇ ⊥ ϼ ∗ r ̇ ⟺ ϼ ∗ ( s ̇ , r ̇ ) = ϼ + ( s ̇ , r ̇ ) ϼ _ ( s ̇ , r ̇ ) = 0. Recall that X ̇ is smooth at s ̇ if and only if ϼ _ ( s ̇ , r ̇ ) = ϼ + ( s ̇ , r ̇ ) for all r ̇ ∈ 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: 6 Suppose ( X ̇ , ‖·‖) represents a real normed vector space, then (i) ϼ ∗ (t ṡ, r ̇ ) = ϼ ∗ (ṡ, t r ̇ )= t 2 ϼ ∗ (ṡ, r ̇ ) for all ṡ, r ̇ ∈ X ̇ and all t ∈ R. (ii) | ϼ ∗ ( s ̇ , r ̇ ) | ≤ ‖ s ̇ ‖ 2 ‖ r ̇ ‖ 2 for all ṡ, r ̇ ∈ X ̇ . (iii) For all vectors except the zero vector, ṡ, r ̇ ∈ X ̇ , if s ̇ ⊥ ϼ ∗ r ̇ , then ṡ and r ̇ are independent in the linear sense. (iv) ϼ ∗ (ṡ, t ṡ + r ̇ ) = t 2 ‖ s ̇ ‖ 4 +2t ‖ s ̇ ‖ 2 ϼ ( ṡ, r ̇ ) + ϼ ∗ (ṡ, r ̇ ) for all ṡ, r ̇ ∈ 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 ( X ̇ , ‖·‖) represents a real normed vector space then ∀ s ̇ , r ̇ ∈ X ̇ , the functional ϼ LM : X ̇ × X ̇ → R which defines as ϼ LM ( s ̇ , r ̇ ) = | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | , We say that ṡ is ϼ LM - orthogonality to r ̇ , we denoted by ṡ ⊥ ϼ LM r ̇ if ϼ LM ( s ̇ , r ̇ ) = 0 . Proposition 2.2: Suppose ( X ̇ , ‖·‖) represents a real normed vector space. Then (a) ϼ LM ( s ̇ , r ̇ ) ≥ 0 for all ṡ, r ̇ ∈ X ̇ . (non-negative) (b) ϼ LM ( s ̇ , 0 ) = ϼ LM ( 0 , r ̇ ) = 0 for all ṡ, r ̇ ∈ X ̇ . (zero property) (c) ϼ LM ( s ̇ , s ̇ ) = ‖ s ̇ ‖ 4 for all ṡ ∈ X ̇ , also ϼ LM ( s ̇ , s ̇ ) = ϼ ∗ ( s ̇ , s ̇ ) (d) ϼ LM ( s ̇ , r ̇ ) = ϼ LM ( r ̇ , s ̇ ) for all ṡ, r ̇ ∈ X ̇ . (symmetry) (e) ϼ LM ( α s ̇ , β r ̇ ) = | α β | 2 ϼ LM ( s ̇ , r ̇ ) for all s ̇ , r ̇ ∈ X ̇ and all α , β ∈ R . (scalar homogeneity) (f ) ϼ LM ( s ̇ , r ̇ ) ≤ ‖ s ̇ ‖ 2 ‖ r ̇ ‖ 2 for all ṡ, r ̇ ∈ X ̇ . (Cauchy schwarz inequality) (g) ϼ LM ( s ̇ , t s ̇ + r ̇ ) = | ( t 2 ‖ s ̇ ‖ 4 + 2 t ‖ s ̇ ‖ 2 ϼ ( s ̇ , r ̇ ) + ϼ ∗ ( s ̇ , r ̇ ) ) ( ϼ ∗ ( t s ̇ + r ̇ , s ̇ ) ) | (h) For all nonzero vectors ṡ, r ̇ ∈ X ̇ , if s ̇ ⊥ ϼ LM r ̇ , then ṡ and r ̇ are linearly independent. ( linear independence ) Proof: (a) We have ϼ LM ( s ̇ , r ̇ ) = | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | since | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | ≥ 0 then | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | ≥ 0, implies ϼ LM ( s ̇ , r ̇ ) ≥ 0. (b) From definition of ϼ LM -orthogonality we have ϼ LM ( s ̇ , 0 ) = | ϼ ∗ ( s ̇ , 0 ) ϼ ∗ ( 0 , s ̇ ) | = | 0.0 | = 0 In the same way, we prove ϼ LM ( 0 , r ̇ ) = 0 . (c) From definition of ϼ ∗ -orthogonality and ϼ LM -orthogonality we get ϼ LM ( s ̇ , s ̇ ) = | ϼ ∗ ( s ̇ , s ̇ ) ϼ ∗ ( s ̇ , s ̇ ) | = | ϼ − ( s ̇ , s ̇ ) ϼ + ( s ̇ , s ̇ ) ϼ − ( s ̇ , s ̇ ) ϼ + ( s ̇ , s ̇ ) | = | ‖ s ̇ ‖ 2 ‖ s ̇ ‖ 2 ‖ s ̇ ‖ 2 ‖ s ̇ ‖ 2 | = ‖ s ̇ ‖ 4 (d) ϼ LM ( s ̇ , r ̇ ) = | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | = | ϼ ∗ ( r ̇ , s ̇ ) ϼ ∗ ( s ̇ , r ̇ ) | = ϼ LM ( r ̇ , s ̇ ) (e) By theorem 1.1 (i) we have ϼ ∗ ( t s ̇ , r ̇ ) = ϼ ∗ ( s ̇ , t r ̇ ) = t 2 ϼ ∗ ( s ̇ , r ̇ ) for all s ̇ , r ̇ ∈ X ̇ and all t ∈ R . ϼ ∗ ( α s ̇ , β r ̇ ) = α 2 β 2 ϼ ∗ ( s ̇ , r ̇ ) and ϼ ∗ ( β r ̇ , α s ̇ ) = α 2 β 2 ϼ ∗ ( r ̇ , s ̇ ) then ϼ LM ( α s ̇ , β r ̇ ) = | ϼ ∗ ( α s ̇ , β r ̇ ) ϼ ∗ ( β r ̇ , α s ̇ ) | = | α 2 β 2 ϼ ∗ ( s ̇ , r ̇ ) α 2 β 2 ϼ ∗ ( r ̇ , s ̇ ) | = | α β | 2 | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | = | α β | 2 ϼ LM ( s ̇ , r ̇ ) (f ) By theorem 1.1 (ii) we have | ϼ ∗ ( s ̇ , r ̇ ) )| ≤ ‖ s ̇ ‖ 2 ‖ r ̇ ‖ 2 for all s ̇ , r ̇ ∈ X ̇ Then ϼ LM ( s ̇ , r ̇ ) = | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | ≤ ‖ s ̇ ‖ 2 ‖ r ̇ ‖ 2 ‖ r ̇ ‖ 2 ‖ s ̇ ‖ 2 = ‖ s ̇ ‖ 2 ‖ r ̇ ‖ 2 for all s ̇ , r ̇ ∈ X ̇ Hence ϼ LM ( s ̇ , r ̇ ) ≤ ‖ s ̇ ‖ 2 ‖ r ̇ ‖ 2 for all ṡ, r ̇ ∈ X ̇ (g) By theorem 1.1 (iv) we have ϼ ∗ ( s ̇ , t s ̇ + r ̇ ) = t 2 ‖ s ̇ ‖ 4 +2t ‖ s ̇ ‖ 2 ϼ ( s ̇ , r ̇ ) + ϼ ∗ ( s ̇ , r ̇ ) ϼ LM ( s ̇ , t s ̇ + r ̇ ) = | ϼ ∗ ( s ̇ , t s ̇ + r ̇ ) ϼ ∗ ( t s ̇ + r ̇ , s ̇ ) | = | ( t 2 ‖ s ̇ ‖ 4 + 2 t ‖ s ̇ ‖ 2 ϼ ( s ̇ , r ̇ ) + ϼ ∗ ( s ̇ , r ̇ ) ) ( ϼ ∗ ( t s ̇ + r ̇ , s ̇ ) ) | (h) we have ṡ ⊥ ϼ LM r ̇ therefore; ϼ LM ( s ̇ , r ̇ ) = | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | = 0 Thus ϼ ∗ ( s ̇ , r ̇ ) = 0 or ϼ ∗ ( r ̇ , s ̇ ) = 0 By theorem 1.1 (iii) for all nonzero vectors ṡ, r ̇ ∈ X ̇ , if s ̇ ⊥ ϼ ∗ r ̇ , then ṡ and r ̇ are linearly independent. Hence, whether ϼ ∗ ( s ̇ , r ̇ ) = 0 or ϼ ∗ ( r ̇ , s ̇ ) = 0 it follows that ṡ and r ̇ are linearly independent. Remark 2.3: If X ̇ come from inner product space then ϼ LM ( s ̇ , r ̇ ) = ϼ ∗ ( s ̇ , r ̇ ) . 3. The connection between ϼ LM – orthogonality and some others orthogonalities Theorem 3.1: Suppose ( X ̇ , ‖ · ‖ ) represents a real normed vector space and let s ̇ , r ̇ ∈ X ̇ . (a) If s ̇ ⊥ ϼ ∗ r ̇ , then s ̇ ⊥ ϼ LM r ̇ (b) If s ̇ ⊥ ϼ − r ̇ , then s ̇ ⊥ ϼ LM r ̇ (c) If s ̇ ⊥ ϼ + r ̇ , then s ̇ ⊥ ϼ LM r ̇ (d) If r ̇ ⊥ ϼ ∗ s ̇ , then s ̇ ⊥ ϼ LM r ̇ (e) If r ̇ ⊥ ϼ − s ̇ , then s ̇ ⊥ ϼ LM r ̇ (f ) If r ̇ ⊥ ϼ + s ̇ , then s ̇ ⊥ ϼ LM r ̇ Proof: Obvious Remark 3.2: The converse of (a), (b) and (c) of theorem 3.1 are not true. For example, take s ̇ = ( 0 , 2 ) and r ̇ = ( 1 , 1 ) if ‖ ( s ̇ 1 , s ̇ 2 ) ‖ = max { | s ̇ 1 | , | s ̇ 2 | } in the space X ̇ = R 2 ϼ + ( s ̇ , r ̇ ) = ‖ s ̇ ‖ lim t → 0 + ‖ s ̇ + t r ̇ ‖ − ‖ s ̇ ‖ t = ( 2 ) lim t → 0 + | 2 + t | − 2 t = 2 ( 1 ) = 2 ≠ 0 ϼ − ( s ̇ , r ̇ ) = ‖ s ̇ ‖ lim t → 0 − ‖ s ̇ + t r ̇ ‖ − ‖ s ̇ ‖ t = ( 2 ) lim t → 0 − | 2 + t | − 2 t = 2 ( 1 ) = 2 ≠ 0 ϼ + ( r ̇ , s ̇ ) = ‖ r ̇ ‖ lim t → 0 + ‖ r ̇ + t s ̇ ‖ − ‖ r ̇ ‖ t = ( 1 ) lim t → 0 + | 1 + 2 t | − 1 t = 1 ( 2 ) = 2 ϼ − ( r ̇ , s ̇ ) = ‖ r ̇ ‖ lim t → 0 − ‖ r ̇ + t s ̇ ‖ − ‖ r ̇ ‖ t = ( 1 ) lim t → 0 − | 1 | − 1 t = Since ϼ ∗ ( s ̇ , r ̇ ) = ϼ + ( s ̇ , r ̇ ) ϼ _ ( s ̇ , r ̇ ) ϼ ∗ ( s ̇ , r ̇ ) = ( 2 ) ( 2 ) = 4 ≠ 0 But ϼ LM ( s ̇ , r ̇ ) = | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | = 0 Hence ṡ ⊥ ϼ LM 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 r ̇ = (1,1,0) if ‖ ( s ̇ 1 , s ̇ 2 , s ̇ 3 ) ‖ = | s ̇ 1 | + | s ̇ 2 | + | s ̇ 3 | in the space X = R 3 , then ϼ + ( s ̇ , r ̇ ) = ‖ s ̇ ‖ lim t → 0 + ‖ s ̇ + t r ̇ ‖ − ‖ s ̇ ‖ t = ( 2 ) lim t → 0 + | 2 + t | + | t | − 2 t = 2 ( 2 ) = 4 ϼ − ( s ̇ , r ̇ ) = ‖ s ̇ ‖ lim t → 0 − ‖ s ̇ + t r ̇ ‖ − ‖ s ̇ ‖ t = ( 2 ) lim t → 0 − | 2 + t | + | t | − 2 t = 2 ( 0 ) = 0 ϼ + ( r ̇ , s ̇ ) = ‖ r ̇ ‖ lim t → 0 + ‖ r ̇ + t s ̇ ‖ − ‖ r ̇ ‖ t = ( 2 ) lim t → 0 + | 1 + 2 t | + | 1 | − 2 t = 2 ( 2 ) = 4 ≠ 0 ϼ − ( r ̇ , s ̇ ) = ‖ r ̇ ‖ lim t → 0 − ‖ r ̇ + t s ̇ ‖ − ‖ r ̇ ‖ t = ( 2 ) lim t → 0 − | 1 + 2 t | + | 1 | − 2 t = 2 ( 2 ) = 4 ≠ 0 Since ϼ ∗ ( r ̇ , s ̇ ) = ϼ + ( r ̇ , s ̇ ) ϼ − ( r ̇ , s ̇ ) ϼ ∗ ( r ̇ , s ̇ ) = ( 4 ) ( 4 ) = 16 ≠ 0 But ϼ LM ( s ̇ , r ̇ ) = | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | = 0 Hence ṡ ⊥ ϼ LM r ̇ but the others are not. Example 3.4: Consider the real normed space X ̇ = R 2 with the norm ‖ s ̇ ‖ = | s ̇ 1 | + | s ̇ 2 | where s ̇ = ( 1 , 0 ) , r ̇ = ( 0 , 1 ) we have ϼ + ( s ̇ , r ̇ ) = ‖ s ̇ ‖ lim t → 0 + ‖ s ̇ + t r ̇ ‖ − ‖ s ̇ ‖ t and ϼ − ( s ̇ , r ̇ ) = ‖ s ̇ ‖ lim t → 0 − ‖ s ̇ + t r ̇ ‖ − ‖ s ̇ ‖ t Thus ϼ + ( s ̇ , r ̇ ) = (1) lim t → 0 + 1 + t − 1 t = 1 ϼ − ( s ̇ , r ̇ ) = ( 1 ) lim t → 0 − 1 − t − 1 t = -1 ϼ + ( r ̇ , s ̇ ) = (1) lim t → 0 + t + 1 − 1 t = 1 ϼ − ( r ̇ , s ̇ ) = ( 1 ) lim t → 0 − − t + 1 − 1 t = -1 We have ϼ ( s ̇ , r ̇ ) = ϼ − ( s ̇ , r ̇ ) + ϼ + ( s ̇ , r ̇ ) 2 therefore ϼ ( s ̇ , r ̇ ) = − 1 + 1 2 = 0, Since ϼ gg = | ϼ ( s ̇ , r ̇ ) | | ϼ ( r ̇ , s ̇ ) | implies ϼ gg ( s ̇ , r ̇ ) = 0 = 0, Also ϼ λ ( s ̇ , r ̇ ) = λ ϼ − ( s ̇ , r ̇ ) + (1- λ) ϼ + ( s ̇ , r ̇ ) ) = λ ( − 1 ) + (1- λ) (1)= 1-2λ put λ = 1 2 then ϼ λ ( s ̇ , r ̇ ) = 0 , But ϼ LM ( s ̇ , r ̇ ) = | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , s ̇ ) | = | 1 | = 1 ≠ 0 Hence ⊥ ϼ ⊈ ⊥ ϼ LM , ⊥ ϼ gg ⊈ ⊥ ϼ LM and ⊥ ϼ λ ⊈ ⊥ ϼ LM . Example 3.5: Consider the real normed space X ̇ = R 2 with the norm ‖ ( s ̇ , r ̇ ) ‖ = max { | s ̇ | , | r ̇ | } where s ̇ = ( 1 , 1 ) , r ̇ = ( 0 , − 1 ) . Then for every s ̇ , r ̇ ∈ X ̇ we have ϼ + ( s ̇ , r ̇ ) = ( 1 ) lim t → 0 + | 1 | − 1 t = 1 ( 0 ) = 0 ϼ − ( s ̇ , r ̇ ) = ( 1 ) lim t → 0 − | 1 − t | − 1 t = 1 ( − 1 ) = − 1 ϼ + ( r ̇ , s ̇ ) = ( 1 ) lim t → 0 + | t | − 1 t = 1 ( 0 ) = 0 ϼ − ( r ̇ , s ̇ ) = ( 1 ) lim t → 0 − | − 1 + t | − 1 t = lim t → 0 − | t − 1 | − 1 t = − 1 We have ϼ ( s ̇ , r ̇ ) = ϼ − ( s ̇ , r ̇ ) + ϼ + ( s ̇ , r ̇ ) 2 therefore ϼ ( s ̇ , r ̇ ) = − 1 + 0 2 = − 1 2 ≠ 0 and ϼ ( r ̇ , s ̇ ) = ϼ − ( r ̇ , s ̇ ) + ϼ + ( r ̇ , s ̇ ) 2 = − 1 + 0 2 = − 1 2 ϼ gg ( s ̇ , r ̇ ) = ( − 1 2 ) ( − 1 2 ) = ( 1 4 ) = 1 2 ≠ 0 ϼ λ ( ṡ , r ̇ ) = λ ϼ − ( s ̇ , r ̇ ) + ( 1 − λ ) ϼ + ( s ̇ , r ̇ ) ) = λ ( − 1 ) + ( 1 − λ ) ( 0 ) = − λ ≠ 0 but ϼ LM (ṡ, r ̇ ) = | ϼ ∗ ( s ̇ , r ̇ ) ϼ ∗ ( r ̇ , 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. Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 31 Dec 2025 ADD YOUR COMMENT Comment 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 Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (1) version 1 Published: 31 Dec 2025, 14:1486 https://doi.org/10.12688/f1000research.174190.1 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. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article 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: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.12688/f1000research.174190.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 31 Dec 2025 Views 0 Cite How to cite this report: Gnanaprakasam AJ. Reviewer Report For: New Kind of orthogonality in Normed spaces by using ϼ * -orthogonality [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.5256/f1000research.192072.r477500 ) The direct URL for this report is: https://f1000research.com/articles/14-1486/v1#referee-response-477500 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 14 May 2026 Arul Joseph Gnanaprakasam , SRM Institute of Science and Technology, Kattankulathur, Tamil Nadu, India Approved VIEWS 0 https://doi.org/10.5256/f1000research.192072.r477500 The authors introduced a new notion of orthogonality based on norm derivatives and established several interesting properties related to this concept. The proposed orthogonality is discussed in comparison with existing notions in the literature, and the paper provides useful ... Continue reading READ ALL The authors introduced a new notion of orthogonality based on norm derivatives and established several interesting properties related to this concept. The proposed orthogonality is discussed in comparison with existing notions in the literature, and the paper provides useful remarks and illustrative examples to clarify the results. The manuscript is mathematically sound and well organized. The topic is relevant to current research in normed spaces and orthogonality theory, and the obtained results may motivate further studies in this direction. Overall, the paper makes a valuable contribution to the subject and is suitable for indexingin its present form. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Partly Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Functional Analysis I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Gnanaprakasam AJ. Reviewer Report For: New Kind of orthogonality in Normed spaces by using ϼ * -orthogonality [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.5256/f1000research.192072.r477500 ) The direct URL for this report is: https://f1000research.com/articles/14-1486/v1#referee-response-477500 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Ghosh S. Reviewer Report For: New Kind of orthogonality in Normed spaces by using ϼ * -orthogonality [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.5256/f1000research.192072.r468033 ) The direct URL for this report is: https://f1000research.com/articles/14-1486/v1#referee-response-468033 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 30 Apr 2026 Souvik Ghosh , Jadavpur University, Kolkata, West Bengal, India Approved VIEWS 0 https://doi.org/10.5256/f1000research.192072.r468033 The authors tried to provide a new orthogonality based on norm derivatives. The orthogonality here mentioned is coarser than other orthogonalities in the literature. Although they showed that it is incomparable with some other orthogonalities followed by some remarks. The ... Continue reading READ ALL The authors tried to provide a new orthogonality based on norm derivatives. The orthogonality here mentioned is coarser than other orthogonalities in the literature. Although they showed that it is incomparable with some other orthogonalities followed by some remarks. The authors may look for the following problems for further exploration. For instance by investigating the quantitative invariant other gap between two different (incomparable) orthogonalities. Also, there is a scope to study the preservation of such orthogonality. Is the work clearly and accurately presented and does it cite the current literature? No Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? No If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Geometry of Banach spaces (Functional Analysis) I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Ghosh S. Reviewer Report For: New Kind of orthogonality in Normed spaces by using ϼ * -orthogonality [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.5256/f1000research.192072.r468033 ) The direct URL for this report is: https://f1000research.com/articles/14-1486/v1#referee-response-468033 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Ishtiaq U. Reviewer Report For: New Kind of orthogonality in Normed spaces by using ϼ * -orthogonality [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.5256/f1000research.192072.r458032 ) The direct URL for this report is: https://f1000research.com/articles/14-1486/v1#referee-response-458032 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 19 Mar 2026 Umar Ishtiaq , University of Management and Technology, Lahore, Pakistan Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.192072.r458032 The authors need to address the following issues: The manuscript contains several typographical errors and formatting inconsistencies, such as repeated sentences in the introduction and missing mathematical symbols in Theorem 1.1. The definition of ... Continue reading READ ALL The authors need to address the following issues: The manuscript contains several typographical errors and formatting inconsistencies, such as repeated sentences in the introduction and missing mathematical symbols in Theorem 1.1. The definition of $\rho_{LM}$-orthogonality in Definition 2.1 is clear, but the motivation for introducing this specific form of orthogonality could be better justified. The proof of Proposition 2.2 contains multiple formatting issues and missing parentheses, making some steps difficult to follow. In Example 3.4, the calculation of $\rho_{LM}(\dot{s},\dot{t})$ appears to contain an error, as the result should be verified more carefully. The references are inconsistently formatted, with some entries missing page numbers or having incomplete information (e.g., references 4, 5, and 6). The paper references figures and tables that are not included in the provided PDF, making it difficult to verify the claims made in examples. The paper lacks a formal conclusion section to summarize the main findings and their significance in the context of existing orthogonality concepts. Is the work clearly and accurately presented and does it cite the current literature? No Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? No Competing Interests: No competing interests were disclosed. Reviewer Expertise: Functional Analysis I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Ishtiaq U. Reviewer Report For: New Kind of orthogonality in Normed spaces by using ϼ * -orthogonality [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.5256/f1000research.192072.r458032 ) The direct URL for this report is: https://f1000research.com/articles/14-1486/v1#referee-response-458032 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 31 Dec 2025 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 3 Version 1 31 Dec 25 read read read Umar Ishtiaq , University of Management and Technology, Lahore, Pakistan Souvik Ghosh , Jadavpur University, Kolkata, India Arul Joseph Gnanaprakasam , SRM Institute of Science and Technology, Kattankulathur, India Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Gnanaprakasam A. This is an open access peer review report 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. 14 May 2026 | for Version 1 Arul Joseph Gnanaprakasam , SRM Institute of Science and Technology, Kattankulathur, Tamil Nadu, India 0 Views copyright © 2026 Gnanaprakasam A. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The authors introduced a new notion of orthogonality based on norm derivatives and established several interesting properties related to this concept. The proposed orthogonality is discussed in comparison with existing notions in the literature, and the paper provides useful remarks and illustrative examples to clarify the results. The manuscript is mathematically sound and well organized. The topic is relevant to current research in normed spaces and orthogonality theory, and the obtained results may motivate further studies in this direction. Overall, the paper makes a valuable contribution to the subject and is suitable for indexingin its present form. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Partly Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Functional Analysis I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Gnanaprakasam AJ. Peer Review Report For: New Kind of orthogonality in Normed spaces by using ϼ * -orthogonality [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.5256/f1000research.192072.r477500) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1486/v1#referee-response-477500 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Ghosh S. This is an open access peer review report 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. 30 Apr 2026 | for Version 1 Souvik Ghosh , Jadavpur University, Kolkata, West Bengal, India 0 Views copyright © 2026 Ghosh S. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The authors tried to provide a new orthogonality based on norm derivatives. The orthogonality here mentioned is coarser than other orthogonalities in the literature. Although they showed that it is incomparable with some other orthogonalities followed by some remarks. The authors may look for the following problems for further exploration. For instance by investigating the quantitative invariant other gap between two different (incomparable) orthogonalities. Also, there is a scope to study the preservation of such orthogonality. Is the work clearly and accurately presented and does it cite the current literature? No Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? No If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Geometry of Banach spaces (Functional Analysis) I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Ghosh S. Peer Review Report For: New Kind of orthogonality in Normed spaces by using ϼ * -orthogonality [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.5256/f1000research.192072.r468033) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1486/v1#referee-response-468033 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Ishtiaq U. This is an open access peer review report 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. 19 Mar 2026 | for Version 1 Umar Ishtiaq , University of Management and Technology, Lahore, Pakistan 0 Views copyright © 2026 Ishtiaq U. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (0) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The authors need to address the following issues: The manuscript contains several typographical errors and formatting inconsistencies, such as repeated sentences in the introduction and missing mathematical symbols in Theorem 1.1. The definition of $\rho_{LM}$-orthogonality in Definition 2.1 is clear, but the motivation for introducing this specific form of orthogonality could be better justified. The proof of Proposition 2.2 contains multiple formatting issues and missing parentheses, making some steps difficult to follow. In Example 3.4, the calculation of $\rho_{LM}(\dot{s},\dot{t})$ appears to contain an error, as the result should be verified more carefully. The references are inconsistently formatted, with some entries missing page numbers or having incomplete information (e.g., references 4, 5, and 6). The paper references figures and tables that are not included in the provided PDF, making it difficult to verify the claims made in examples. The paper lacks a formal conclusion section to summarize the main findings and their significance in the context of existing orthogonality concepts. Is the work clearly and accurately presented and does it cite the current literature? No Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? No Competing Interests No competing interests were disclosed. Reviewer Expertise Functional Analysis I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. reply Respond to this report Responses (0) Ishtiaq U. Peer Review Report For: New Kind of orthogonality in Normed spaces by using ϼ * -orthogonality [version 1; peer review: 2 approved, 1 approved with reservations] . F1000Research 2025, 14 :1486 ( https://doi.org/10.5256/f1000research.192072.r458032) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-1486/v1#referee-response-458032 Alongside their report, reviewers assign a status to the article: Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions Adjust parameters to alter display View on desktop for interactive features Includes Interactive Elements View on desktop for interactive features Competing Interests Policy Provide sufficient details of any financial or non-financial competing interests to enable users to assess whether your comments might lead a reasonable person to question your impartiality. 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Register $(document).ready(function () { signIn.createSignInAsRow($("#sign-in-form-gfb-popup")); $(".target-field").each(function () { var uris = $(this).val().split("/"); if (uris.pop() === "login") { $(this).val(uris.toString().replace(",","/")); } }); });