Separable Spaces and Alaoglu's Theorem Review

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 $X^*$.
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  • 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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