Countable norming family
= Countable norming family
{title2=$\lVert x\rVert=\sup_n\varphi_n(x)$}
A real <separable Banach space> has a sequence $(\varphi_n)$ in its dual unit ball with $\lVert x\rVert=\sup_n\varphi_n(x)$ for every $x$. Choose a dense sequence $(u_n)$ in the unit sphere and apply the <Hahn-Banach theorem> to obtain $\varphi_n(u_n)=1$. In the complex case use $\sup_n\operatorname{Re}\varphi_n(x)$.