Characterization of Convergent Net in Weak Topology

Theorem

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

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

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

Let $x \in X$.

Let $\struct {\Lambda, \preceq}$ be a directed set.

Let $\family {x_\lambda}_{\lambda \in \Lambda}$ be a net in $X$.

Then:

$\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$ in $\struct {X, w}$

if and only if:

for each $f \in X^\ast$, the net $\family {\map f {x_\lambda} }_{\lambda \in \Lambda}$ converges to $\map f x$.

Proof

Necessary Condition

Suppose that:

$\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$ in $\struct {X, w}$.

Let $f \in X^\ast$.

From Characterization of Continuity of Linear Functional in Weak Topology, $f : \struct {X, w} \to \GF$ is continuous.

From Characterization of Continuity in terms of Nets, we have that:

for each $f \in X^\ast$, the net $\family {\map f {x_\lambda} }_{\lambda \in \Lambda}$ converges to $\map f x$.

$\Box$

Sufficient Condition

Suppose that:

for each $f \in X^\ast$, the net $\family {\map f {x_\lambda} }_{\lambda \in \Lambda}$ converges to $\map f x$.

Let $U$ be an open neighborhood of $x$ in $\struct {X, w}$.

From Open Sets in Weak Topology of Topological Vector Space, there exists $f_1, f_2, \ldots, f_n \in X^\ast$ and $\epsilon > 0$ such that:

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

Then for each $1 \le i \le n$, we have that:

$\family {\map {f_i} {x_\lambda} }_{\lambda \in \Lambda}$ converges to $\map {f_i} x$.

So for each $1 \le i \le n$ there exists $\lambda_i \in \Lambda$ such that for all $\lambda \in \Lambda$ with $\lambda_i \preceq \lambda$ we have:

$\cmod {\map {f_i} {x - x_\lambda} } < \epsilon$

from Characterization of Convergent Net in Metric Space.

From Existence of Upper Bound of Finite Subset of Directed Set, there exists $\lambda_\ast \in \Lambda$ such that $\lambda_i \preceq \lambda_\ast$ for each $1 \le i \le n$.

Then, for $\lambda \in \Lambda$ with $\lambda_\ast \preceq \lambda$, we have $\lambda_i \preceq \lambda$ for each $1 \le i \le n$ by transitivity.

Then:

$\cmod {\map {f_i} {x - x_\lambda} } < \epsilon$

for each $1 \le i \le n$ and $\lambda_\ast \preceq \lambda$.

So $x_\lambda \in U$ for $\lambda_\ast \preceq \lambda$.

So $\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$ in $\struct {X, w}$.

$\blacksquare$