 Separability of the continuous dual implies separability

If the continuous dual space $X^{\prime }$ of a normed vector space is norm separable, then $X$ is separable. Choose a dense sequence $f_n$ in the dual unit ball and unit-ball points $x_n$ with $|f_n(x_n)|\ge \Vert f_n\Vert /2$. If their closed linear span were proper, the Hahn-Banach distance formula would give a norm-one functional $f$ vanishing on it. An $f_n$ within $1/4$ of $f$ would simultaneously satisfy $|f_n(x_n)|<1/4$ and $|f_n(x_n)|>3/8$, a contradiction. Rational linear combinations give a countable dense subset of $X$.

The converse fails by duality of l1 and l infinity: ${\ell }^1$ is separable, whereas ${\ell }^{\infty }$ contains uncountably many zero-one sequences at mutual distance one.