Vanishing sequences norm the summable sequence space (${\sup }_{y\in c_0,\Vert y{\Vert }_{\infty }\le 1}|⟨x,y⟩|=\Vert x{\Vert }_1$)

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.