This is mostly for my own reference, since this fact seems to come up a lot.
Theorem 1 Let be an open connected set and an integral operator with kernel , that is,
Then is compact.
Proof: Let be an orthonormal basis of , then write the kernel as
Define the sequence of integral operators on by
It is clear that , so the are of finite rank and hence compact. For , we have
As , it follows that
We conclude that is compact as the limit of a sequence of compact operators in the operator norm topology.