The images of orthogonally additive mappings from inner product spaces to groups

Abstract

We investigate orthogonally additive mappings $f:E\rightarrow G$ from a real inner product space $(E,||\cdot ||)$ of dimension at least 2 to a group $(G,\cdot )$ and show that there are additive functions $a:{\mathbb {R}}\rightarrow G$ and $b:E\rightarrow G$ such that $f(x)=a(||x||^2)b(x)$ for all $x\in E$. Moreover the subgroup generated by the image of f is an abelian group, which is n-divisible for every positive integer n.