In Ledoux and Talagrand's "Probability in Banach Spaces", for technical reasons they frequently assume that a Banach space $B$ has the property that the unit ball of $B^*$ contains a countable subset $D$ such that $$ \Vert x \Vert = \sup_{f\in D} \vert f(x) \vert$$ for every $x\in B$. Examples of such spaces $B$ include both all separable Banach spaces, and all duals of separable Banach spaces (e.g. $\ell_\infty$).

My question is, is there a standard name or alternative characterization of spaces with this property? The authors decline to discuss this at all, except to point out that separable spaces and $\ell_\infty$ have this property; they don't even point out that it extends to duals of separable spaces.