I must be missing something basic and simple: If $X$ is a normed vector space and the closed unit ball in $X^{*}$ is weak* compact, and translations and dilations are homeomorphisms, why isn't $X^{*}$ locally compact?
2026-03-30 00:21:10.1774830070
Why doesn't Alaoglu's theorem imply that $X^{*}$ is locally compact in the weak* topology?
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Because a locally compact Hausdorff TVS is finite dimensional and thus carries its usual topology. See for example this blog post by Terry Tao.
More concretely for the specific case of $X^*$: The reason that $X^*$ fails to be weak-$*$ locally compact is that the closed unit ball isn't a neighbourhood of $0$ in the weak-$*$ topology.