= Vanishing sequences norm the summable sequence space
{title2=$\sup_{y\in c_0,\ \|y\|_\infty\leq1}|\langle x,y\rangle|=\|x\|_1$}
Finite truncations of the sign or phase sequence of $x\in\ell^1$ belong to $c_0$ and attain increasing partial sums of $\sum_n|x_n|$. Thus the <space of sequences converging to zero> is 1-norming for the <absolutely summable sequence space>. Its codimension in $\ell^\infty$ is infinite: indicators of pairwise disjoint infinite subsets have linearly independent classes modulo $c_0$. Norming does not require finite codimension.
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