Let $V$ be a real Hilbert space. Suppose $a:V\times V\to\mathbb{R}$ is bilinear, continuous, and coercive: there exists $\alpha>0$ such that $a(v,v)\ge \alpha\|v\|_V^2$ for all $v\in V$. Then for every bounded linear functional $F\in V^\*$ there exists a unique $u\in V$ such that $a(u,v)=F(v)$ for all $v\in V$.