Stability of the Generalized Orthogonality Functional Equation

  • Donald H. Hyers
  • George Isac
  • Themistocles M. Rassias
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)

Abstract

Let (E,.) denote a real Hilbert space of dimension greater than one, and let T: E → E be a mapping which satisfies the functional equation
$$\left| {T\left( x \right) \cdot T\left( y \right)} \right| = \left| {x \cdot y} \right|$$
(9.1)
known as the generalized orthogonality equation. We will follow the work of C. Alsina and J.L. Garcia-Roig (1991) (see also Th.M. Rassias (1997)). It is easy to see that any solution of (9.1) will satisfy the following conditions for x, y in E and real µ:
  1. (A)

    ||T(x)|| =||x||.

     
  2. (B)

    T(x) = 0 if and only if x = 0.

     
  3. (C)

    T(x) • T(y) = 0 if and only if xy = 0.

     
  4. (D)

    I cos A(x, y) I = I cos A(T (x), T (y))1, where A(u, v) denotes the angle between the vectors u and v.

     
  5. (E)

    T(µx) = ±µT(x).

     

Keywords

Manifold Hull 

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Donald H. Hyers
  • George Isac
    • 1
  • Themistocles M. Rassias
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

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