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Let $B$ and $C$ be abelian groups (in additive notation). We call a function $f:B\rightarrow C$ a quadratic form if for all $x,y,z \in B$, the function $f$ satisfies the relation $$f(x+y+z)-f(x+y)-f(y+z)-f(x+z)+f(x)+f(y)+f(z)=0.$$

Is there some reference which says that it satisfies the parallelogram law $f(x+y)+f(x-y)=2f(x)+2f(y)$ and that for $B=\mathbb{Z}^n$ we can write $$f(\displaystyle\sum_{i=1}^{n} a_i e_i)=\displaystyle\sum_{i=1}^n\big(2a_i^2+\displaystyle\sum_{j=1}^n a_ia_j \big) f(e_i)+ \displaystyle\sum_{1\leq i where $e_i$ denote the standard basis vectors of $\mathbb{Z}^n$.

Thank you!

  • 1
    Do you specifically need a reference? How about just a proof?2012-01-13
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    Do you have one? I do not see how to do it.2012-01-13

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