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Because $w_k\ge c>0$, a finite sum ${\sum }_k{w}_k|u_k|$ implies $u\in {\ell }^1$. Thus the printed weighted one-norm on l2 equals the extended nonnegative sum on every $u\in {\ell }^2$. Unbounded weights may make that sum infinite for some $u\in {\ell }^1$; membership in ${\ell }^1$ alone is not a finiteness assertion.

For each $N$, $J_N(u)={\sum }_{k=1}^N{w}_k|u_k|$ is continuous in the l2 sequence space, and $J={\sup }_N{J}_N$. If $u^{(n)}\to u$ in norm, then for every $N$,Taking the supremum over $N$ provesIt is also weakly lower semicontinuous by the same finite-coordinate argument, and convex and proper because zero and finite sequences have finite penalty.

At a finite-penalty base point $v$, coordinate variations give the subdifferential of a weighted one-norm on l2:These conditions are sufficient by summing the coordinate supporting inequalities. Since $w_k\ge c$, an infinitely supported $v$ would require infinitely many $|p_k|\ge c$, contradicting $p\in {\ell }^2$. Thus a finite-penalty point can have empty subdifferential; a Bregman base with an actual subgradient must be finitely supported.

For any such $p$, $⟨p,v⟩=J(v)$ and the Bregman distance simplifies toEach summand is nonnegative; the sum may be infinite. The pairing converges by Cauchy-Schwarz inequality. This definition does not silently assume a subgradient exists at every $v\in {\ell }^1$.