= Weighted one-norm on l2
{title2=$J_w(u)=\sum_k w_k|u_k|$}
For $0<c\leq w_k<\infty$, 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>.
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