[3] The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces.
Specifically, let L2(X) be a separable Hilbert space and X a locally compact Hausdorff space equipped with a positive Borel measure.
The initial condition on the kernel k on Ω ⊆ Rn can be reinterpreted as demanding k belong to L2(X × X).
If then T is also self-adjoint and so the spectral theorem applies.
This is one of the fundamental constructions of such operators, which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces.