Let $A$ be an abelian group. $A$ function $d : A → R$ is a quadratic form if it satisfies the following conditions: (i) $d(α)=d(−α)$ for all $α∈A$. (ii) The pairing $A × A → R$ , $( α , β ) → d ( α + β ) − d ( α ) − d ( β )$ , is bilinear. A quadratic form $d$ is positive definite if it further satisfies: (iii)$ d(α)≥0$ for all $α∈A$. (iv) $d(α)=0$ ifandonlyif $α=0$.
My question:
Let $d:A→\mathbb{Z}$ be a positive definite quadratic form, and let $n$ be integer, and $φ∈A$, then does $d(mφ)＝m^2d(φ)$ hold?
This problem appeared in the process of proving causy Schwarz inequality of positive difinitequadratic version.