Hahn-Banach Separation Theorem/Normed Vector Space/Real Case

Theorem
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\R$.

Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $\struct {X, \norm \cdot}$.

Also known as
These theorems are sometimes known as the geometric Hahn-Banach theorems.