Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Separable
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Theorem
Let $X$ be a separable normed vector space.
Let $X^\ast$ be the normed dual space of $X$.
Let $B^-_{X^\ast}$ be the closed unit ball of $X^\ast$.
Let $w^\ast$ be the weak-$\ast$ topology on $B^-_{X^\ast}$.
Then $\struct {B^-_{X^\ast}, w^\ast}$ is separable.
Proof
From the Banach-Alaoglu Theorem, $\struct {B^-_{X^\ast}, w^\ast}$ is compact.
From Closed Unit Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable, $\struct {B^-_{X^\ast}, w^\ast}$ is metrizable.
From Compact Metric Space is Separable, we have that $\struct {B^-_{X^\ast}, w^\ast}$ is separable.
$\blacksquare$