Annihilator of Subspace of Banach Space is Weak-* Closed

Theorem

Let $X$ be a Banach space.

Let $M$ be a vector subspace of $X$.

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

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

Let $M^\bot$ be the annihilator of $M$.

Then $M^\bot$ is closed in $\struct {X^\ast, w^\ast}$.

$M^\bot = \map {\cl_{w^\ast} } {M^\bot}$

$M^\bot \subseteq \map {\cl_{w^\ast} } {M^\bot}$

Now let:

$f \in \map {\cl_{w^\ast} } {M^\bot}$

From Point in Set Closure iff Limit of Net, there exists a directed set $\struct {\Lambda, \preceq}$ and a net $\family {f_\lambda}_{\lambda \in \Lambda}$ in $M^\bot$ converging to $f$ in $\struct {X^\ast, w^\ast}$.

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

Since $f_\lambda \in M^\bot$ for each $\lambda \in \Lambda$, we have:

$\map {f_\lambda} x = 0$ for each $x \in M$ and $\lambda \in \Lambda$.

the net $\family {\map {f_\lambda} x}_{\lambda \in \Lambda}$ converges to $\map f x$ and $0$ in $\GF$.

Hence from Characterization of Hausdorff Property in terms of Nets, we obtain $\map f x = 0$ for each $x \in M$.

So:

$f \in M^\bot$

So we obtain:

$\map {\cl_{w^\ast} } {M^\bot} \subseteq M^\bot$

and hence:

$M^\bot = \map {\cl_{w^\ast} } {M^\bot}$

From Set is Closed iff Equals Topological Closure, we have that $M^\bot$ is closed in $\struct {X^\ast, w^\ast}$.

$\blacksquare$

$\ds M^\bot = \bigcap_{x \in M} \map \ker {x^\wedge}$

the linear functional $x^\wedge : \struct {X^\ast, w^\ast} \to \GF$ is continuous.

So $M^\bot$ is the intersection of closed sets in $\struct {X^\ast, w^\ast}$, and hence is closed itself.

$\blacksquare$