If $P_n=\{y\in M(n,\Bbb R)|$ $y$ is positive definite and symmetric $\}$ then for a fixed $y \in P_n$ consider the set $A=\{y[u]|u \in \Bbb Z^n\}$ where $y[u]=u^tyu$ is clearly positive. Now, how to prove that $\exists u_1\in P_n$ s.t $y[u_1]$ is minimal element of the set $A$ and there are finit...

I am facing problem to understand Minkowski's reduction theory from Klingen's Siegel Modular form book. I am giving the pictures of the corresponding pages. I am stuck there for hours. $P_n=\{y\in M(n,\Bbb R)| y>0\}$ Now my question is "What does minimal mean where he states "$y[u_1]$ be...