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

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.