Separable Criterion: Weak-* Com./Seq. Comp. of Closed Unit Ball of X*

# Separable Criterion for the Weak-* Compactness and Weak-* Sequential Compactness of the Closed Unit Ball of X*

Recall from the Helley's Theorem page that Helley's theorem states that if $X$ is a normed linear space and $X$ is separable then every bounded sequence in $X^*$ will weak-* converge.

We will now use Helley's theorem to show that if $X$ is a normed linear space and $X$ is separable then the closed unit ball of $X^*$ is weak-* compact and weak-* sequentially compact.

 Theorem 1: Let $X$ be a normed linear space. If $X$ is separable then the closed unit ball of the topological dual $X^*$ is compact and sequentially compact in the weak-* topology.

In general the concepts of compactness and sequential compactness are different. If $X$ is a metric space, they are the same.

• Proof: Let $B_{X^*}$ denote the closed unit ball of $X^*$. That is:
(1)
\begin{align} \quad B_{X^*} = \{ f \in X^* : \| f \| \leq 1 \} \end{align}