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Study on Orthogonal Sets for Birkhoff Orthogonality

Author

Listed:
  • Xiaomei Wang

    (Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China)

  • Donghai Ji

    (Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China)

  • Yueyue Wei

    (Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China)

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. We prove that the Birkhoff orthogonal set A = { x 1 , x 2 , … , x n } ( n ≥ 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.

Suggested Citation

  • Xiaomei Wang & Donghai Ji & Yueyue Wei, 2023. "Study on Orthogonal Sets for Birkhoff Orthogonality," Mathematics, MDPI, vol. 11(20), pages 1-11, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4320-:d:1261431
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    References listed on IDEAS

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    1. Alexander A. Katz, 2020. "A Note on Symmetry of Birkhoff-James Orthogonality in Positive Cones of Locally C *-algebras," Mathematics, MDPI, vol. 8(6), pages 1-14, June.
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