On the Continuity of Orthogonal Sets in the Sense of Operator Orthogonality


  • Mahdi Iranmanesh Shahrood uninersity of technology
  • M. Saeedi Khojasteh
  • M. K. Anwary




Birkhoff orthogonality‎, ‎set valued function‎, ‎upper semi continuous‎, ‎lower semi continuous


In this paper, we introduce the operator approach for orthogonality in linear spaces. In particular, we represent the concept of orthogonal vectors using an operator associated with them, in normed spaces. Moreover, we investigate some of continuity properties of this kind of orthogonality. More precisely, we show that the set valued function F(x; y) = {μ : μ ∈ C, p(x − μy, y) = 1} is upper and lower semi continuous, where p(x, y) = sup{pz1,...,zn−2 (x, y) : z1, . . . , zn−2 ∈ X} and pz1,...,zn−2 (x, y) = kPx,z1,...,zn−2,yk−1 where Px,z1,...,zn−2,y denotes the projection parallel to y from X to the subspace generated by {x, z1, . . . , zn−2}. This can be considered as an alternative definition for numerical range in linear spaces.

Author Biography

  • Mahdi Iranmanesh, Shahrood uninersity of technology
    Department of Mathematics, Shahrood University of Technology, Iran






Mathematics of Finance

How to Cite

On the Continuity of Orthogonal Sets in the Sense of Operator Orthogonality. (2018). European Journal of Pure and Applied Mathematics, 11(3), 793-802. https://doi.org/10.29020/nybg.ejpam.v11i3.3237

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