Separation in dual space Let $X$ be a real Banach space and $X^*$ its dual space.
Let $C^*$ be a weak$^*$ closed and convex subset in $X^*$ 
and $x^*\notin C^*$. Then there exists $x\in X$ such that
$$
\langle x^*, x\rangle > \sup_{f\in C^*}\langle f, x\rangle.
$$
I would like to ask whether the statement is true? How can we prove?
 A: This is just the separation theorem (see e.g. Rudin, "Functional Analysis", Theorem 3.4(b)) for the locally convex topological vector space $X^*$ with the weak-* topology (whose continuous linear functionals correspond to the points of $X$): slightly more generally you can replace $x^*$ by a weak-* compact
convex set disjoint from $C^*$. 
A: I don't have the full answer as of now, but the best that I can think of is that if $ X $ is a reflexive Banach space, then the statement is true by the geometric Hahn-Banach Theorem. You can treat $ X^{*} $ as the initial Banach space under consideration and $ C^{*} $ as the closed and convex (and non-empty!) subset of $ X^{*} $. Then if $ x^{*} \notin C^{*} $, one can find an $ x \in X \cong X^{**} $ (by the geometric Hahn-Banach Theorem) such that
\begin{equation}
\langle x^{*},x \rangle > \sup_{f \in C^{*}} \langle f,x \rangle. 
\end{equation}
Now, there is an error associated with the phrasing of the problem. As you are taking the supremum of the right-hand side of the inequality over $ f \in C^{*} $, there is no need to universally quantify $ f $. On a separate note, if $ C^{*} $ is weak*-closed, then it is automatically closed, so you do not have to add the adjective 'closed' in front of 'weak*-closed'.
If $ X $ is not reflexive, then I think the statement is false. That is because the separating functional that you need to separate $ x^{*} $ and $ C^{*} $ might lie in $ X^{**} \setminus X $. However, I could be wrong, so any opposing opinions are welcome.
Addendum
The above discussion is concerned with the case when $ C^{*} $ is closed with respect to the norm topology on $ X^{*} $. After the latest edit of the problem statement, it is now understood that $ C^{*} $ is to be assumed closed with respect to the weak*-topology on $ X^{*} $. More generally, observe that if we assume $ C^{*} $ to be closed with respect to any locally convex topology $ \mathcal{T} $ that is finer than the weak*-topology (denoted by $ \sigma(X^{*},X) $) but coarser than the Mackey topology (denoted by $ \tau(X^{*},X) $), then the continuous dual of $ (X^{*},\mathcal{T}) $ is still isomorphic to the natural copy of $ X $ in $ X^{**} $. Hence, the separation theorem mentioned by Robert (I call it the geometric Hahn-Banach Theorem) still applies to yield a separating functional $ x \in X $.
