Open Sets in Weak Topology of Topological Vector Space

Theorem

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

Let $X$ be a topological vector space over $\GF$ with weak topology $w$.

Let $U \subseteq X$.

Then $U$ is open in $\struct {X, w}$ if and only if for each $x \in U$ there exists $f_1, f_2, \ldots, f_n \in X^\ast$ and $\epsilon > 0$ such that:

$\set {y \in X : \cmod {\map {f_i} {y - x} } < \epsilon \text { for each } 1 \le i \le n} \subseteq U$

Proof

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

By the definition of the weak topology, $w$ is the initial topology on $X$ with respect to $X^\ast$.

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

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

for each $x \in X$, and set:

$\PP = \set {p_f : f \in X^\ast}$

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

From Open Sets in Standard Topology of Locally Convex Space, we obtain that $U$ is open in $\struct {X, w}$ if and only if for each $x \in U$ there exists $f_1, f_2, \ldots, f_n \in X^\ast$ and $\epsilon > 0$ such that:

$\set {y \in X : \map {p_{f_i} } {y - x} < \epsilon \text { for each } 1 \le i \le n} = \set {y \in X : \cmod {\map {f_i} {y - x} } < \epsilon \text { for each } 1 \le i \le n} \subseteq U$

$\blacksquare$