How does Schwartz kernel theorem on manifold hold

Question

For the Schwartz kernel theorem on $R^n$, the topology of $S^\prime$ is the strong one(limit topology of a directed system of Sobolev spaces, Weak convergence in the space of tempered distributions and weighted Sobolev spaces).

But on a noncompact manifold, the Sobolev spaces may not be well defined. So in this case how is the Schwartz kernel theorem proved?