Normed Dual Space Separates Points
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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