Equivalence of norms in finite dimension

Statement

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$. Then any two norms on $V$ are equivalent.

Remarks

Proof idea: fix a basis and compare a norm $\|\cdot\|$ with the sup norm $\|\cdot\|_\infty$ of the coordinates. The triangle inequality gives $\|x\| \le C \|x\|_\infty$, so $\|\cdot\|$ is continuous for $\|\cdot\|_\infty$. The unit sphere of $\|\cdot\|_\infty$ is compact by the Heine–Borel theorem, so $\|\cdot\|$ has a positive minimum $c$ on it, by the extreme value theorem. Thus there is only one norm topology on $\mathbb{R}^n$, the Euclidean topology. The statement fails in infinite dimensions: each infinite-dimensional normed space has norms that are not equivalent.

Explanations

L1 human

In finite dimensions, it does not matter which norm you choose: all norms give the same convergent sequences and the same open sets.