Given a normed separable vector space $X$, the Banach-Alaoglu theorem states that the closed unit ball in the dual space $X^*$ is compact in the weak-* topology. Since $X$ is separable, that topology is metrizable.
Now, the Alaoglu theorem states that for any normed vector space, the closed unit ball is compact iff the space is finite dimensional. Applying this to $X^*$ allegedly yields that $X^*$ is finite dimensional (obviously false).
Where is my mistake?