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14 pages, 304 KB

Open AccessArticle

Trapezoid Orthogonality in Complex Normed Linear Spaces

by Zheng Li, Tie Zhang and Changjun Li

Mathematics 2025, 13(9), 1494; https://doi.org/10.3390/math13091494 - 30 Apr 2025

Viewed by 282

Let

G_{p} (x, y, z) = {∥ x + y + z ∥}^{p} + {∥ z ∥}^{p} {- ∥ x + z ∥}^{p} - {∥ y + z ∥}^{p}

be defined on a normed [...] Read more.

Let

G_{p} (x, y, z) = {∥ x + y + z ∥}^{p} + {∥ z ∥}^{p} {- ∥ x + z ∥}^{p} - {∥ y + z ∥}^{p}

be defined on a normed space

X

. The special case

G_{2} (x, y, z) = 0, \forall z \in X

, where

X

is a real normed linear space, coincides with the trapezoid orthogonality (T-orthogonality), which was originally proposed by Alsina et al. in 1999. In this paper, for the case where

X

is a complex inner product space endowed with the inner product

⟨ \cdot, \cdot ⟩

and induced norm

∥ \cdot ∥

, it is proved that

S g n (G_{2} (x, y, z)) = S g n (R e ⟨ x, y ⟩), \forall z \in X

, and a geometric explanation for condition

R e ⟨ x, y ⟩ = 0

is provided. Furthermore, a condition

G_{2} (x, i y, z) = 0, \forall z \in X

is added to extend the T-orthogonality to the general complex normed linear spaces. Based on some characterizations, the T-orthogonality is compared with several other well-known types of orthogonality. The fact that T-orthogonality implies Roberts orthogonality is also revealed. Full article

11 pages, 241 KB

Open AccessArticle

Relationship between Generalized Orthogonality and Gâteaux Derivative

by Peixuan Xu, Donghai Ji and Hongxu Zhang

Mathematics 2024, 12(3), 364; https://doi.org/10.3390/math12030364 - 23 Jan 2024

Viewed by 1060

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

⊥_{Δ}

coordinate system in a smooth Minkowski plane. Relevant examples are given at the end of the paper. Full article

(This article belongs to the Section B: Geometry and Topology)

11 pages, 257 KB

Open AccessArticle

Study on Orthogonal Sets for Birkhoff Orthogonality

by Xiaomei Wang, Donghai Ji and Yueyue Wei

Mathematics 2023, 11(20), 4320; https://doi.org/10.3390/math11204320 - 17 Oct 2023

Viewed by 1198

Abstract

We introduce the notion of orthogonal sets for Birkhoff orthogonality, which we will call Birkhoff orthogonal sets in this paper. As a generalization of orthogonal sets in Hilbert spaces, Birkhoff orthogonal sets are not necessarily linearly independent sets in finite-dimensional real normed spaces. [...] Read more.

A = {x_{1}, x_{2}, \dots, x_{n}} (n \geq 3)

containing

n - 3

right symmetric points is linearly independent in smooth normed spaces. In particular, we obtain similar results in strictly convex normed spaces when

n = 3

and in both smooth and strictly convex normed spaces when

n = 4

. These obtained results can be applied to the mutually Birkhoff orthogonal sets studied in recently. Full article

(This article belongs to the Section B: Geometry and Topology)

14 pages, 294 KB

Open AccessArticle

A Note on Symmetry of Birkhoff-James Orthogonality in Positive Cones of Locally C*-algebras

by Alexander A. Katz

Mathematics 2020, 8(6), 1027; https://doi.org/10.3390/math8061027 - 23 Jun 2020

Cited by 1 | Viewed by 2282

Abstract

In the present note some results of Kimuro, Saito, and Tanaka on symmetry of Birkhoff-James orthogonality in positive cones of C*-algebras are extended to locally C*-algebras. Full article

(This article belongs to the Special Issue Noncommutative Geometry and Number Theory)

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