Relationship between Generalized Orthogonality and Gâteaux Derivative

作者：Peixuan Xu, Donghai Ji, Hongxu Zhang · 发表于：Mathematics · 年份：2024 · DOI：10.3390/math12030364 · 研究领域：Advanced Optimization Algorithms Research、Functional Equations Stability Results、Matrix Theory and Algorithms

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.