Fredholm's theorem

In mathematics, Fredholm's theorems are a set of celebrated results of Ivar Fredholm in the Fredholm theory of integral equations.

There are several closely related theorems, which may be stated in terms of integral equations, in terms of linear algebra, or in terms of the Fredholm operator on Banach spaces.

Fredholm's theorem in linear algebra is as follows: if M is a matrix, then the orthogonal complement of the row space of M is the null space of M: Similarly, the orthogonal complement of the column space of M is the null space of the adjoint: Fredholm's theorem for integral equations is expressed as follows.

be an integral kernel, and consider the homogeneous equations and its complex adjoint Here,

Then, Fredholm's theorem is that, for any fixed value of

, these equations have either the trivial solution

or have the same number of linearly independent solutions

A sufficient condition for this theorem to hold is for

to be square integrable on the rectangle

In Fredholm theory, this result generalizes to integral operators on multi-dimensional spaces, including, for example, Riemannian manifolds.

One of Fredholm's theorems, closely related to the Fredholm alternative, concerns the existence of solutions to the inhomogeneous Fredholm equation Solutions to this equation exist if and only if the function

is orthogonal to the complete set of solutions

of the corresponding homogeneous adjoint equation: where

and the former is one of the complete set of solutions to A sufficient condition for this theorem to hold is for

to be square integrable on the rectangle