The outline presented here is broad, whereas the difficulty of formalizing this sketch is, of course, in the details.
It is sometimes called the nucleus of the integral, whence the term nuclear operator arises.
In the general theory, x and y may be points on any manifold; the real number line or m-dimensional Euclidean space in the simplest cases.
Examples of such spaces are the orthogonal polynomials that occur as the solutions to a class of second-order ordinary differential equations.
That this is the same kernel as before follows from the completeness of the basis of the Hilbert space, namely, that one has Since the ωn are generally increasing, the resulting eigenvalues of the operator K(x,y) are thus seen to be decreasing towards zero.
The zeta function plays an important role in studying dynamical systems.
One of the important results from the general theory is that the kernel is a compact operator when the space of functions are equicontinuous.
Fredholm's 1903 paper in Acta Mathematica is considered to be one of the major landmarks in the establishment of operator theory.