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Relationship between Generalized Orthogonality and Gâteaux Derivative

Author

Listed:
  • Peixuan Xu

    (Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China
    Heilongjiang Provincial Key Laboratory of Optimization Control and Intelligent Analysis for Complex Systems, Harbin University of Science and Technology, Harbin 150080, China)

  • Donghai Ji

    (Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China
    Heilongjiang Provincial Key Laboratory of Optimization Control and Intelligent Analysis for Complex Systems, Harbin University of Science and Technology, Harbin 150080, China)

  • Hongxu Zhang

    (Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China
    Heilongjiang Provincial Key Laboratory of Optimization Control and Intelligent Analysis for Complex Systems, Harbin University of Science and Technology, Harbin 150080, China
    Heilongjiang Provincial Key Laboratory of Complex Intelligent System and Integration, Harbin University of Science and Technology, Harbin 150080, China)

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.

Suggested Citation

  • Peixuan Xu & Donghai Ji & Hongxu Zhang, 2024. "Relationship between Generalized Orthogonality and Gâteaux Derivative," Mathematics, MDPI, vol. 12(3), pages 1-11, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:3:p:364-:d:1324780
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