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.

$\blacksquare$