A characterization of generalized inner product spaces (ϕ-orthogonally additive mappings, III)

Abstract

The conditional Cauchy functional equation for a mappingF: (X, +, ⊥) → (Y, +), i.e.,F(x + y) = F(x) + F(y) for allx, y ∈ X withx ⊥ y, (*) on a real vector space equipped with an abstract relation ⊥ (calledorthogonality), was first studied by Gudder and Strawther in 1975. They defined ⊥ by a system consisting of five axioms and described the general hemi-continuous real valued solution of (*) showing that the existence of non-trivial even ones characterize inner product orthogonality. Using the more restrictive axioms of Ratz (introduced in 1980 to obtain the general solution without regularity conditions: odd solutions are additive, while the even ones are quadratic), recently we have proved the same assuming arbitrary mappingsF with values in an abelian group but for dimX ⩾ 3. In 1989, Ratz and the author modified the system of axioms so that it should include the orthogonality induced by an isotropic symmetric bilinear form ϕ and still ensure the additive/quadratic representation. In this context, the main purpose of this note is to characterize on a real vector space the symmetric bilinear orthogonality ⊥ϕ as the essentially unique extension of an orthogonality relation satisfying certain weak axioms and admitting non-trivial even hemi-continuous solutions of (*) with values in a Hausdorff topological abelian group.