Recently I have been doing some readings on topological vector spaces (TVS) and I have some problems reconciliating two seemingly related but different concepts.
The first concept is that of a locally compact topological space: a topological space in which every point has a compact neighborhood. The second is that of a TVS with the Heine-Borel property: a TVS in which every closed bounded subset is compact. At first sight I thought that the Heine-Borel property may imply locally compactness, but this seems to be not true.
I read in several places that one can prove that every locally compact Hausdorff TVS has finite dimension. However there are infinite dimensional Hausdorff TVS with the Heine-Borel property. In particular infinite dimensional Fréchet–Montel spaces, such as the Schwartz space $\mathcal{S}(\mathbb{R}^n)$, are Hausdorff (included in the Fréchet part) and have the Heine-Borel property (from the Montel part). But by the result previously mentioned, these infinite dimensional Fréchet–Montel spaces cannot be locally compact. How can this be possible? The only way I could think is if there is a point in these spaces without a compact neighborhood. But the topology of Fréchet spaces can be generated by a translation invariant metric, which makes every point topologically similar to any other point. Which means, for infinite dimensional Fréchet–Montel spaces, that no point has a compact neighborhood. But from the Heine-Borel property we have that any close bounded subset is a compact neighborhood of any point in its interior. I am certainly confused here. What am I missing?
As Paul Frost indicates in his comment, for infinite dimensional Fréchet–Montel spaces every closed bounded set has to have an empty interior. If this were not true, there would be a closed bounded set which will be a compact neighborhood of any point in its interior. Because for Fréchet spaces every point is topologically similar to any other point, that will imply that every point will have a compact neighborhood, i.e. the space will be locally compact, which is impossible for infinite dimensional Hausdorff TVS.