A metric space $X$ is said to have the Heine-Borel property if every closed and bounded set is compact. There are two well-known facts about this property:
- Every Euclidean space $\mathbb{R}^n$ adopts the Heine-Borel property.
- No infinite dimensional Banach space adopts the Heine-Borel property.
What is the intuition behind the difference between these two types of metric spaces? So far I've mostly dealt with Euclidean spaces, thus have a tendency to associate compactness being the same as closed and bounded. What are the heuristics behind this property failing for infinite dimensional cases?
A set in a metric space is compact iff it is complete and totally bounded. Here totally bounded means that for any $\varepsilon>0$, there exist finitely many open balls of radius $\varepsilon$ that cover the set. In $\mathbb R^n$, every bounded set is totally bounded (this would be a good exercise to prove).
In infinite dimensional Banach spaces, this is no longer the case. Take $S = \{e_n : n=1,2,\ldots\}\subset \ell^\infty(\mathbb R)$, where $e_n$ is the $n^{\mathrm{th}}$ standard orthonormal basis vector (when viewed in $\ell^2$). Then clearly $S$ is bounded as $\|x\|_\infty = 1$ for all $x\in S$, but for all $x,y\in S$ we have $d(x,y)=1$, so for $\varepsilon=1$ there can be no finite collection of $\varepsilon$ balls that cover the set. Hence $S$ is not totally bounded.