Annihilator of Subspace of Banach Space is Weak-* Closed/Proof 2

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 $\struct {X^\ast, w^\ast}$.

Proof

From Annihilator of Subspace of Banach Space as Intersection of Kernels, we have:

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

From Characterization of Continuity of Linear Functional in Weak-* Topology:

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

From Characterization of Continuous Linear Functionals on Topological Vector Space, $\map \ker {x^\wedge}$ is closed in $\struct {X^\ast, w^\ast}$.

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

$\blacksquare$