Are finite dimensional normed linear spaces locally compact?

Question

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.

Answer 1

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.

Answer 2

If $X$ is a finite dimensional vector space, then there exists a unique linear topology on $X$. It follows that if you choose a linear isomorphism $T \ : \ X \to \mathbb R^n$, then $T$ is a homeomorphism. Since $\mathbb R^n$ is locally compact, so is $X$.