Orthogonality of polynomials and orthosymmetry (Q2343349)

Given vector spaces \(X\) and \(Y\), a function \(P: X\to Y\) is said to be an \(n\)-homogeneous polynomial if there is an \(n\)-linear mapping \(\Psi: \underbrace{X\times\cdots\times X}_{\text{n-times}}\to Y\) such that \(P(x)=\Psi(x, \dots,x)\) for all \(x\) in \(X\). If \(A\) and \(B\) are Archimedean vector lattices, an \(n\)-homogeneous polynomial \(P: A\to B\) is said to be orthogonally additive if \(P(x+y)=P(x)+P(y)\) for all \(x,y\) in \(A\) with \(|x|\wedge|y|=0\), while an \(n\)-linear mapping \(\Psi: {A\times\cdots\times A}\to B\) is said to be orthosymmetric if \(\Psi(a_1,\dots,a_n)=0\) if \(a_i\wedge a_j=0\) for some \(1\leq i,j\leq n\). A vector lattice \(A\) is said to be hyper-Archimedean if \(A/I\) is Archimedean for every order ideal \(I\) in \(A\). In this paper, the author uses \(\Omega\) to denote the linear mapping which associates with each \(n\)-linear mapping, \(\Psi: A\times\dots\times A\to {\mathbb R}\), the \(n\)-homogeneous polynomial, \(P=\Omega(\Psi): A\to {\mathbb R}\), defined by \(P(x)=\Psi(x,\dots, x)\). He shows that \(\Omega\) is an isomorphism from the set of all orthosymmetric \(n\)-linear mappings on \(A\times\ldots\times A\) onto the set of all orthogonally additive polynomials on \(A\) if and only if \(A\) is hyper-Archimedean.