Open Sets in Weak-* Topology of Topological Vector Space

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 $X^\ast$.

Let $U \subseteq X$.

Then $U$ is open in $\struct {X^\ast, w^\ast}$ if and only if for each $f \in X^\ast$ there exists $x_1, x_2, \ldots, x_n \in X$ and $\epsilon > 0$ such that:

$\set {g \in X^\ast : \cmod {\map f {x_i} - \map g {x_i} } < \epsilon \text { for each } 1 \le i \le n}$

Proof

For each $x \in X$, define $p_x : X^\ast \to \hointr 0 \infty$ by:

$\map {p_x} f = \cmod {\map f x}$

for each $f \in X^\ast$, and set:

$\PP = \set {p_x : x \in X}$

From the definition of the weak-$\ast$ topology, $w^\ast$ is generated by $\set {x^\wedge : x \in X}$.

So from from Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex, $w^\ast$ is the standard topology on the locally convex space $\struct {X^\ast, \PP}$.

From Open Sets in Standard Topology of Locally Convex Space, we obtain that $U$ is open in $\struct {X^\ast, w^\ast}$ if and only if for each $f \in X^\ast$ there exists $x_1, x_2, \ldots, x_n \in X$ and $\epsilon > 0$ such that:

$\set {g \in X^\ast : \cmod {\map f {x_i} - \map g {x_i} } < \epsilon \text { for each } 1 \le i \le n}$

$\blacksquare$