Countable norming family ($\Vert x\Vert ={\sup }_n{\phi }_n(x)$)

A real separable Banach space has a sequence $({\phi }_n)$ in its dual unit ball with $\Vert x\Vert ={\sup }_n{\phi }_n(x)$ for every $x$. Choose a dense sequence $(u_n)$ in the unit sphere and apply the Hahn-Banach theorem to obtain ${\phi }_n(u_n)=1$. In the complex case use ${\sup }_n\mathrm{Re}{\phi }_n(x)$.