Weak-* Topology is Hausdorff
Theorem
Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological vector space over $\GF$.
Let $X^\ast$ be the topological dual space of $X$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
Then $\struct {X^\ast, w^\ast}$ is Hausdorff space.
Proof
From Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex, $w^\ast$ can be induced by:
- $\PP = \set {p_{x^\wedge} : x \in X}$
where we define $p_{x^\wedge} : X^\ast \to \R_{\ge 0}$ by:
- $\map {p_{x^\wedge} } f = \cmod {\map {x^\wedge} f} = \cmod {\map f x}$
From Locally Convex Space is Hausdorff iff induces Hausdorff Topology, it suffices to show that $\struct {X^\ast, w^\ast}$ is Hausdorff as a locally convex space.
That is, that $\PP$ is separating.
Let $f \in X^\ast$ be such that $f \ne 0$.
Then there exists $x \in X$ such that $\map f x \ne 0$.
Then $\map {p_{x^\wedge} } f = \map {x^\wedge} f \ne 0$.
So $\PP$ is indeed separating.
$\blacksquare$