Compact Operators

Let $T:V\to W$ be a linear map between normed spaces it is called a compact operator* if for every bounded sequence $(v_k)$ in $V$, the sequence $(T{v}_k)$ contains a convergent subsequence in $W$.

Another description is that $T$ is a compact operator if, every bounded subset of $V$ is mapped into a relatively compact set in $W$. This actually means that every compact operator is a bounded one.

Lastly, if the image of the unit ball of $V$ under $T$ is relatively compact in $W$