On orthogonality spaces admitting nontrivial even orthogonally additive mappings (Q756027)

Let (X,\(\perp)\) be an orthogonality space (X a real vector space and \(\perp\) an orthogonality relation on X). Let \((Y,+)\) be an abelian group. Let \((e)Hom_{\perp}(X,Y)\) denote the set of mappings \(\{\) E: \(X\to Y|\) E is even orthogonally additive\(\}\), i.e., \(E(x+y)=E(x)+E(y)\) for all x,y\(\in X\) with \(x\perp y\), E even. The author proves: If dim \(X\geq 3\) and \((e)Hom_{\perp}(X,Y)\neq \{0\}\), then there exists a functional \(\rho\) : \(X\to {\mathbb{R}}\) with the following properties: (i) \(\rho (0)=0\), \(\rho (x)>0\) for all \(x\in X\setminus \{0\}\); (ii) \(\rho (\lambda x)=| \lambda | \rho (x)\), for all \(x\in X\) and \(\lambda\in {\mathbb{R}}\); (iii) \(E(x)=E(y)\) for any \(E\in (e)Hom_{\perp}(X,Y)\) and every x,y\(\in X\) such that \(\rho (x)=\rho (y).\) The main result: If dim \(X\geq 3\) and \((e)Hom_{\perp}(X,Y)\neq \{0\}\), then there exists an equivalent inner product \(<\cdot,\cdot >: X\times Y\to {\mathbb{R}}\), i.e. \(x\perp y\Leftrightarrow <x,y>=0\) for all x,y\(\in X\), and \((e)Hom_{\perp}(X,Y)=\{a\circ \| \cdot \|^ 2|\) \(a\in Hom({\mathbb{R}},Y)\}\).