Let $V$ be a topological $\mathbb{R}$-vector space with ${\rm dim}(V) < \infty$
Then, $V$ is locally compact $??$
2026-03-25 11:32:37.1774438357
A linear topological space over real number field is locally compact ???
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A standard theorem in topological vector spaces (over the reals) says that if $n = \dim(V) < \infty$ then $V \simeq \mathbb{R}^n$, so in particular it is locally compact. This is non-trivial, and the converse also holds: if the dimension is not finite, then $V$ is not locally compact.
E.g. See theorems 1.20 and 1.21 in Rudin's Functional Analysis. Or pages 244-245 of Dunford and Schwartz Linear Operators Part I, General Theory. To name the first two books on functional analysis on my shelf I tried. It's in most standard texts.