Sequential Banach-Alaoglu theorem for a separable predual
= Sequential Banach-Alaoglu theorem for a separable predual
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If a <normed vector space> $X$ is separable, every bounded sequence in $X^*$ has a subsequence that converges pointwise on $X$. Equivalently, the closed unit ball of $X^*$ is sequentially compact in the <weak-star topology>. A <diagonal subsequence argument> on a countable dense subset proves the result.