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, *

*
* where

* Then is compact. *

*Proof:* Let be an orthonormal basis of , then write the kernel as

where

Define the sequence of integral operators on by

where

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.

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