Heine–Borel theorem in finite-dimensional normed spaces

Statement

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, with a norm $\|\cdot\|$ and the metric $d(x, y) = \|x - y\|$. Then a subset $K \subseteq V$ is compact if and only if $K$ is closed and bounded.

Remarks

Proof: choose a basis. By equivalence of norms in finite dimension, the norm is equivalent to the Euclidean norm of the coordinates. Equivalent norms have the same closed sets, bounded sets and compact sets. Then use the Heine–Borel theorem in $\mathbb{R}^n$ or $\mathbb{C}^n = \mathbb{R}^{2n}$. In other words, each finite-dimensional normed space is a proper metric space. The converse also holds: if the closed unit ball is compact, the space is finite-dimensional.

Explanations

L1 human

In each finite-dimensional normed space, the compact sets are the closed bounded sets, just as in $\mathbb{R}^n$.