Weighted one-norm on l2 ($J_w(u)={\sum }_k{w}_k|u_k|$)

For $0<c\le w_k<\infty$, interpret the nonnegative sum as an extended-valued functional on the l2 sequence space. Finiteness implies $u\in {\ell }^1$, but unbounded weights can make it infinite even there. It is proper and convex. Every finite partial sum is norm-continuous, so their supremum is sequentially lower semicontinuous and also weakly lower semicontinuous. This is useful as a sparsity penalty in variational regularization.

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 $w_k\ge c>0$, a subgradient can exist only at finitely supported $u$. At such a base point $v$, $D_{J_w}^p(u,v)={\sum }_k(w_k|u_k|-p_k{u}_k)$, allowing the value infinity. Merely having finite penalty does not guarantee a nonempty subdifferential in the ambient l2 sequence space.