 Uniqueness for a strongly monotone variational inequality

Let $K$ be a convex set in a real Banach space, and let $A:K\to X^{\ast }$ be a strongly monotone operator. If $u,w\in K$ both satisfy the displayed variational inequality, testing with $v=w$ and $v=u$ respectively and adding yields $⟨Au-Aw,u-w⟩\le 0$. Strong monotonicity gives $m\Vert u-w{\Vert }^2\le 0$, hence $u=w$. This proves uniqueness whenever a solution exists.