Relationship between Generalized Orthogonality and Gâteaux Derivative

Peixuan Xu, Donghai Ji and Hongxu Zhang ()
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Peixuan Xu: Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China
Donghai Ji: Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China
Hongxu Zhang: Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China

Mathematics, 2024, vol. 12, issue 3, 1-11

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.

Keywords: Gâteaux derivative; isosceles orthogonality; Birkhoff orthogonality; approximate orthogonality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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