Let $V$ be a real normed finite dimensional vector space. Is it true that $A \subseteq V$ compact iff $A$ is closed and bounded?

Question

Let $V$ be a real normed finite dimensional vector space. Is it true that $A \subseteq V$ compact iff $A$ is closed and bounded?

I know the result is true if the vector space is $\mathbb{R}^n$ itself. To prove this, I should prove that isomorphisms map bounded sets to bounded sets and closed sets to closed sets, but for some reason, this is hard for me.

Is a stronger statement true? I.e., do linear isomorphisms preserve compactness?

Answer 1

All linear transformations on a finite dimensional space are continuous, and any finite dimensional space is isomorphic to $\mathbb{R}^n$.

Answer 2

On a finite-dimensional real vector space, all norms are equivalent. So, if your statement is true for the usual norm, it is true for every norm.