Let $X$ be a real [Banach space](/page/Banach%20Space), let $K\subset X$ be a nonempty convex set, and let $I:X\to\mathbb R$ be Gateaux differentiable. If $u\in K$ minimizes $I$ over $K$, then
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\begin{align*}
I'[u](v-u)\ge 0
\end{align*}
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for every $v\in K$.
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Conversely, if $I$ is convex and $u\in K$ satisfies
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\begin{align*}
I'[u](v-u)\ge 0
\end{align*}
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for every $v\in K$, then $u$ minimizes $I$ over $K$.