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If $B$ is a quadratic form over some space $V$, what is the norm determined by $B$? Is this the inner product $\langle Bu,Bv\rangle$?

If not, and it is not possible to determine a norm from knowing its quadratic form, is there another way to evaluate $\Vert u\Vert$?

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Actually, this seems to be a bit mixed up. If $q\colon V\to F$ is a quadratic form (and the characteristic of $F$ is $\ne 2$), then $B\colon V\times V\to F$, $(v,w)\mapsto \frac{q(v+w)-q(v)-q(w)}2$ is a symmetric bilinear form. Note that $B(v,v)=q(v)$. If moreover, $F\subseteq \mathbb R$ and the forms are postive definite (i.e. $B$ is a scalar product), then $\lVert\cdot\rVert\colon V\to\mathbb R$, $v\mapsto \sqrt{q(v)}=\sqrt{B(v,v)}$ is a norm.

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