Heine-Borel Theorem/Normed Vector Space/Necessary Condition

Theorem

Let $\struct {X, \norm {\, \cdot \,} }$ be a finite-dimensional normed vector space.

Let $K \subseteq X$ be closed and bounded.

Then $K$ is a compact subset.

Proof

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $K$.

$\sequence {x_n}_{n \mathop \in \N}$ is bounded.

We have that bounded sequence in finite-dimensional space has a convergent subsequence.

Hence, $\sequence {x_n}_{n \mathop \in \N}$ has a convergent subsequence $\sequence {x_{n_k} }_{k \mathop \in \N}$.

Denote the limit $\ds \lim_{k \mathop \to \infty} \sequence {x_{n_k} } = L$.

By assumption, $K$ is closed.

By equivalence of definitions, $L \in K$.

By definition, $K$ is compact.

$\blacksquare$

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 1.5$: Normed and Banach spaces. Compact sets