Are finite dimensional normed linear spaces locally compact? > > Let $(X,\| \cdot \|)$ be a finite dimensional normed linear space. Can we say that $(X,\| \cdot \|)$ is a locally compact normed linear space? Please help me in understanding this concept. Thank you very much.

If $X$ is a normed vector space on $\mathbb{R}$ (or $\mathbb{C}$) then $X$ is locally compact iff it is finite dimensionnal.

$\mathbb{Q}$, as seen as a $1$-dimensionnal normed vector space on $\mathbb{Q}$, is not locally compact since no neighbourhood of the origin is compact.

Proving this is the same as proving that the closed unit ball of $X$ is compact iff $X$ is finite dimensionnal. You can find a proof or hints here.