Minty's lemma ($⟨Au,v-u⟩\ge 0⟺⟨Av,v-u⟩\ge 0$)

For a monotone operator $A:K\to X^{\ast }$ on a nonempty convex set $K$ in a real Banach space, assume hemicontinuity. Then $u\in K$ solves $⟨Au,v-u⟩\ge 0$ for every $v\in K$ if and only if $⟨Av,v-u⟩\ge 0$ for every $v\in K$. The quantifier applies to the entire inequality on each side.

The forward implication is operator monotonicity. For the reverse implication, choose $v=u+t(w-u)$ with $w\in K$ and $t>0$. Divide the assumed inequality by $t$ and pass to $t\downarrow 0$ using hemicontinuity; the result is $⟨Au,w-u⟩\ge 0$. For $K=X$, applying this to $A-f$ shows that $Au=f$ is equivalent to $⟨Av-f,v-u⟩\ge 0$ for every $v\in X$.