Let $X$ be a locally convex topological vector space with topology $\tau$. Let $X^*$ be the set of linear functionals on $X$ endowed with the weak* topology. Let $Z^* \subset X^*$ be a vector subspace such that the topology $Z^*$ induces on $X$ is equal to $\tau$ when restricted to some subset $D$ of $X$.
Suppose $x\in D\subset X$ and $C$ is an $X$-closed subset of $D\subset X$ with $x\notin C$. Is it true that there exists a linear functional $\alpha \in Z^*$ on $D$ that strictly separates $x$ from $C$?
For example, $X$ might be the set of bounded measures on a sigma algebra, $Z^*$ might be the set of finitely-valued, measurable functions, and $D$ the set of probability measures on that sigma algebra.
I'd be happy to just get a reference. Thanks
