Let $(X,\Vert\cdot\Vert)$ be a normed $\mathbb{K}$-vector space and $A \subset X$ be closed and bounded. My problem is how to determine whether $A$ is compact?
I know that a compact subset is always closed and bounded. And that in $\mathbb{R}$ the converse holds due to Bolzano–Weierstraß.
But I think I am missing something. Do there exist subsets in normed vector spaces which are closed and bounded but not compact?
Best Answer
Yes.
The equivalence of "compact" and "closed and bounded" holds only in finite-dimensional spaces.
In a general, infinite-dimensional normed space (e.g., over the real scalars) complete in its norm (i.e., an infinite-dimensional Banach space), for a set to be compact, it is not enough that it be closed and bounded. It must be closed and totally bounded.