User:Caliburn/s/nets/Weak-* Dense Subset of Topological Dual Space Separates Points

Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \tau}$ be a topological vector space over $\GF$.

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

Suppose that $X^\ast$ separates the points of $X$.

Let $w^\ast$ be the weak-$\ast$ topology on $X$.

Let $S$ be an everywhere dense subset of $\struct {X^\ast, w^\ast}$.

Then $S$ separates points.

That is, for each $x, y \in X$ with $x \ne y$ there exists $f \in S$ with:

$\map f x \ne \map f y$

Proof

Suppose that $x, y \in X$ have $\map f x = \map f y$ for each $f \in S$.

Let $g \in X^\ast$.

From Point is in Topological Closure iff Limit of Moore-Smith Sequence, there exists a directed set $\struct {\Lambda, \preceq}$ and a Moore-Smith sequence $\family {f_\lambda}_{\lambda \in \Lambda}$ in $S$ such that $\family {f_\lambda}_{\lambda \in \Lambda}$ converges to $g$.

We have:

$\map {f_\lambda} x = \map {f_\lambda} y$ for each $\lambda \in \Lambda$.

From Characterization of Convergent Moore-Smith Sequences in Weak-* Topology, we have that:

the Moore-Smith sequences $\family {\map {f_\lambda} x}_{\lambda \in \Lambda}$ converges to $\map g x$ and $\map g y$.

From Characterization of Hausdorff Spaces in terms Moore-Smith Sequences and Weak-* Topology is Hausdorff, we have that:

$\map g x = \map g y$

for all $g \in X^\ast$.

Since $X^\ast$ separates the points of $X$, we have $x = y$.