= Subdifferential of a weighted one-norm on l2
{title2=$p_k=w_k\operatorname{sign}(u_k)\ (u_k\ne0),\quad |p_k|\leq w_k\ (u_k=0)$}
The coordinate conditions define a <subgradient> precisely when $p\in\ell^2$ and $J_w(u)<\infty$. They follow by varying one coordinate; summing their scalar supporting inequalities proves sufficiency. Since $w_k\geq c>0$, a <subgradient> can exist only at finitely supported $u$. At such a base point $v$, $D_{J_w}^p(u,v)=\sum_k(w_k|u_k|-p_ku_k)$, allowing the value infinity. Merely having finite penalty does not guarantee a nonempty <subdifferential> in the ambient <l2 sequence space>.
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