Characterization of Convergent Net in Weak-* Topology

Theorem

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

Let $\struct {X, \tau}$ be a topological vector space.

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

Let $w^\ast$ be the weak-$\ast$ topology on $X$.

Let $f \in X^\ast$.

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

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

Then:

$\family {f_\lambda}_{\lambda \in \Lambda}$ converges to $f$ in $\struct {X^\ast, w^\ast}$

if and only if:

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

Proof

Necessary Condition

Suppose that:

$\family {f_\lambda}_{\lambda \in \Lambda}$ converges to $f$ in $\struct {X^\ast, w^\ast}$

Let $x \in X$.

From Characterization of Continuity of Linear Functional in Weak-* Topology, $x^\wedge : \struct {X^\ast, w^\ast} \to \GF$ is continuous.

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

for each $f \in X^\ast$, the net $\family {\map {x^\wedge} {f_\lambda} }_{\lambda \in \Lambda} = \family {\map {f_\lambda} x}_{\lambda \in \Lambda}$ 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 $f$ in $\struct {X^\ast, w^\ast}$.

From Open Sets in Weak-* Topology of Topological Vector Space, there exists $x_1, \ldots, x_n \in X$ and $\epsilon > 0$ such that:

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

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

$\family {\map {f_\lambda} {x_i} }_{\lambda \in \Lambda}$ converges to $\map f {x_i}$.

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:

$\cmod {\map {f_\lambda} {x_i} - \map f {x_i} } < \epsilon$

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:

$\cmod {\map {f_\lambda} {x_i} - \map f {x_i} } < \epsilon$

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

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

So $\family {f_\lambda}_{\lambda \in \Lambda}$ converges to $f$ in $\struct {X^\ast, w^\ast}$.

$\blacksquare$