Partial-sum norm on absolutely summable sequences ($q(x)={\sup }_k|{\sum }_{n\le k}{x}_n|$)

On ${\ell }^1$, $q$ is a norm, bounded above by $\Vert x{\Vert }_1$. The partial-sum map $Jx=({\sum }_{n\le k}{x}_n{)}_k$ is an isometric linear embedding into the space $c$ of convergent sequences with the supremum norm. Its range contains every eventually constant sequence: take successive differences of such a sequence to obtain a finitely supported element of ${\ell }^1$. These sequences are dense in $c$, so the completion of a normed space is $c$.

The range is proper: partial sums of the alternating harmonic series converge but their successive differences do not belong to ${\ell }^1$. The finite truncations of these differences are $q$-Cauchy and cannot have a limit in ${\ell }^1$. Thus this normed version of ${\ell }^1$ is incomplete, despite the completeness of its usual norm.