Closed Unit Ball in Normed Dual Space is Weak-* Closed

Theorem

Let $X$ be a normed vector space.

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

Let $B^-_{X^\ast}$ be the closed unit ball in $X^\ast$.

Then we have that $B^-_{X^\ast}$ is weak-$\ast$ closed.

Proof

From Weak-* Topology is Hausdorff, $\struct {X^\ast, w^\ast}$ is Hausdorff.

From the Banach-Alaoglu Theorem, $\struct {B^-_{X^\ast}, w^\ast}$ is compact.

From Compact Subspace of Hausdorff Space is Closed, it follows that $B^-_{X^\ast}$ is weak-$\ast$ closed.

$\blacksquare$