Closed Convex Set in terms of Bounded Linear Functionals

Theorem
Let $X$ be a normed vector space over $\R$.

Let $X^\ast$ be the normed dual of $X$.

Let $C$ be a closed convex subset of $X$.

Then:


 * $\ds C = \bigcap_{f \in X^\ast} \set {x \in X : \map f x \le \sup_{c \in C} \map f c}$