In mathematics, the Fredholm integral equation introduced by Ivar Fredholm gives rises to a Fredholm operator. From the point of view of functional analysis it therefore has a well-understood abstract eigenvalue theory. In this case that is supported by a computational theory, including the Fredholm determinants.

Writing K for the integration operator determined by the continuous kernel function K(x,y) on an interval [a,b] of the real line, a Fredholm integral operator is of type

I + K,

and the typical integral equation to solve is of the inhomogeneous type

(I + K)f = g

where g is a given function on [a,b]. Here K is a compact operator (to show this one relies on equicontinuity). It therefore has a spectral theory that can be understood in terms of a discrete spectrum of eigenvalues that tend to 0. This underlies the theory of the equation.

See also: Liouville-Neumann series.