In the highlighted sentence, $ K_n$ is a closed subset of $V^o$, which is the polar of $V$ and so is compact in the weak-* topology by the Banach-Alaoglu theorem. Therefore, $K_n$ is weak-* compact. Theorem 1.3.13 refers to the Baire category theorem. Since $K_n$ is locally compact, we only need that $K_n$ is Hausdorff. Here, how can I know that $K_n$ is Hausdorff so that I may apply the category theorem? Or, is there something that I misunderstood? Thanks.
The weak$^\ast$ topology is Hausdorff and so are all its subspaces. So it's no issue.