 Weighted one-norm on l2

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.