The Schwartz kernel theorem works for operators defined on $C_ {c}(\mathbb{R}^n,E)$, as long as $E$ is finite-dimensional and we introduce the right notion of a generalised section. In every construction I know of, we make explicit use of the fact that $E$ is canonically isomorphic to $E^{**}$. This makes it problematic (or simply false, I expect) to extend it to an arbitary infinite-dimensional case.
Nevertheless, I'd like to know if we can make some statement similar to the Schwartz kernel theorem, for example for "nice" Frechet spaces. Can we naively adopt the theorem for reflexive spaces? Do we have some characterisation regarding the kind of operator we can represent by a kernel even if the full theorem does not hold?
Thank you!