= Minty's lemma
{c}
{title2=$\langle Au,v-u\rangle\geq0\Longleftrightarrow\langle Av,v-u\rangle\geq0$}
For a <monotone operator> $A:K\to X^*$ on a nonempty <convex set> $K$ in a real <Banach space>, assume <hemicontinuity>. Then $u\in K$ solves $\langle Au,v-u\rangle\geq0$ for every $v\in K$ if and only if $\langle Av,v-u\rangle\geq0$ 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 $t\downarrow0$ using <hemicontinuity>; the result is $\langle Au,w-u\rangle\geq0$. For $K=X$, applying this to $A-f$ shows that $Au=f$ is equivalent to $\langle Av-f,v-u\rangle\geq0$ for every $v\in X$.
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