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Results in Mathematics

, Volume 30, Issue 1–2, pp 25–38 | Cite as

On functional which are orthogonally additive modulo Z

  • Janusz Brzdȩk
Article

Abstract

Let E be a real inner product space with dimension at least 2, D ⊂ E, f: E → R with f(x+y)−f(x)−f(y) ∈ Z for all orthogonal x,y ∈ E, and f(D) ⊂ (−γ,γ)+Z witn some real γ > 0. We prove that, under some additional assumptions, there are a unique linear functional A: E → R and a unique constant d ∈ R with f(x)−d∥x∥2−A(x) ∈ Z for x ∈ E. We also show some applications of this result to the determination of solutions F: E → C of the conditional equation: F(x+y) = F(x)F(y) for all orthogonal x,y ∈ E.

Mathematics Subject Classification (1991)

39B52 

Keywords

Cauchy difference Christensen measurability Baire property inner product space 

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Copyright information

© Birkhäuser Verlag, Basel 1996

Authors and Affiliations

  • Janusz Brzdȩk
    • 1
  1. 1.Department of MathematicsPedagogical UniversityRzeszówPoland

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