TITLE: Banach spaces with a certain separability property
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: A subset D of $B^{*}$ with the property you describe is called "norming" (wether or not it is countable).
Recall that a subset D is called "total" if if $d(x) = 0$ for all $d \in D$ implies $x = 0$.
Every norming set is total.
Maybe a partial answer to your question is given by proposition 5.17 in the "Handbook of the Geometry of Banach Spaces" volume 1, chapter 15 "Infinite dimensional convexity", it says when the converse is true:
Every total subset of $B^{*}$ 
is norming iff B is quasireflexive, that is $\frac{dim(B^{**})}{dim B} \lt \infty$.