Separable Spaces and Alaoglu's Theorem Review
 Table of Contents

# Separable Spaces and Alaoglu's Theorem Review

We will now review some of the recent material regarding separable spaces and Alaoglu's theorem.

• On the Helly's Theorem page we said that a topological space \$X\$ is said to be Separable if \$X\$ contains a countable and dense subset.
• We then proved Helly's theorem, which states that if \$X\$ is a separable normed linear space then every bounded sequence of continuous linear functionals in \$X^*\$ has a subsequence that weak* converges in \$X^*\$.
• We then proved Alaoglu's Theorem which states that if \$X\$ is a normed linear space then the closed unit ball in \$X^*\$ is compact with respect to the weak* topology.
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