= Quadratic energy minimization by the parallelogram law
{title2=$Q(v_j-v_k)=2Q(v_j)+2Q(v_k)-4Q((v_j+v_k)/2)$}
Let $V$ be a closed <linear subspace> of a real <Hilbert space>, and let $Q(v)=a(v,v)$ for a symmetric <bounded bilinear form> that is nonnegative on the ambient space and coercive on $V$. Then $Q$ has a unique minimizer on every closed <affine subspace> $g+V$. If $Q(v_j)\to m=\inf_{g+V}Q$, the midpoint belongs to $g+V$ and the <parallelogram law> gives $Q(v_j-v_k)\leq2Q(v_j)+2Q(v_k)-4m\to0$. Coercivity on $V$ makes the <minimizing sequence> Cauchy; completeness and continuity give a minimizer. The vanishing <first variation> is $a(u,v)=0$ for all $v\in V$. Two minimizers differ by $w\in V$ with $Q(w)=0$, so coercivity gives uniqueness. This proves the symmetric existence theorem directly, without invoking the <Lax-Milgram theorem>.
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