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 $X$.

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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

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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:

$x = y$

$\blacksquare$

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $20.1$: Existence of a Support Functional