 Norm attainment on the vanishing sequence space

The continuous linear functional $f_a(x)=\sum a_n{x}_n$ on the real space of sequences converging to zero corresponds to $a\in {\ell }_1$ and has operator norm $\sum |a_n|$. Equality $f_a(x)=\Vert f_a\Vert$ on the closed unit ball forces $x_n=\mathrm{sgn}(a_n)$ on the support of $a$, since every term $|a_n|-a_n{x}_n$ is nonnegative. Such an $x$ can tend to zero only if that support is finite. Conversely, the finite sign sequence attains equality for every finitely supported sequence $a$.