= Subgradient inversion under convex conjugacy
{title2=$p\in\partial J(u)\iff u\in\partial J^*(p)$}
For a proper lower-semicontinuous convex functional on a real <Hilbert space>, $p\in\partial J(u)$ means $\langle v,p\rangle-J(v)\leq\langle u,p\rangle-J(u)$ for all $v$. Taking the supremum gives equality in the <Fenchel–Young inequality>: $J(u)+J^*(p)=\langle u,p\rangle$. Apply the same argument to $J^*$ and use the <Fenchel-Moreau theorem> $J^{**}=J$ to obtain the reciprocal <subgradient> condition.
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