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)
Cauchy difference Christensen measurability Baire property inner product space
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