Normed Dual Space Separates Points

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

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

Then $X^\ast$ separates points.

That is, suppose that $x, y \in X$ are such that:


 * $\map f x = \map f y$ for each $f \in X^\ast$.

Then $x = y$.

Proof
From Existence of Support Functional, there exists a $\phi \in X^\ast$ such that:


 * $\map \phi {x - y} = \norm {x - y}$

Since $\phi$ is linear, we then have:


 * $\map \phi x - \map \phi y = \norm {x - y}$

By hypothesis, we have:


 * $\map \phi x = \map \phi y$

so:


 * $\norm {x - y} = 0$

Since a norm is positive definite, we then have:


 * $x - y = 0$

so that:


 * $x = y$