Wikpedia says that if $X,Y$ are normed spaces and $T: X \to Y$ is a linear operator, then the following two statements are equivalent:
- $T$ is a compact operator
- for any bounded sequence $(x_n)_{n \in \mathbb{N}}$ in $X$, the sequence $(T x_n)_{n \in \mathbb{N}}$ contains a converging subsequence.
To give a counter example, the identity map on $\mathbb{Q}$ is compact. But there are bounded sequences of rational numbers with Cauchy subsequences but not convergent subsequences.
If $Y$ is complete, and therefore a Banach space, then all Cauchy sequences converge. But Wikipedia clearly says this is for an NLS, not necessarily a complete NLS (Banach).
Is my counter-example wrong or is Wikipedia wrong?