= Partial-sum norm on absolutely summable sequences
{title2=$q(x)=\sup_k|\sum_{n\leq k}x_n|$}
On $\ell^1$, $q$ is a <norm>, bounded above by $\|x\|_1$. The partial-sum map $Jx=(\sum_{n\leq k}x_n)_k$ 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.
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