# Fredholm alternative

In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.

## Linear algebra

If V is an n-dimensional vector space and $T:V\to V$ is a linear transformation, then exactly one of the following holds:

1. For each vector v in V there is a vector u in V so that $T(u)=v$ . In other words: T is surjective (and so also bijective, since V is finite-dimensional).
2. $\dim(\ker(T))>0$ .

A more elementary formulation, in terms of matrices, is as follows. Given an m×n matrix A and a m×1 column vector b, exactly one of the following must hold:

1. Either: A x = b has a solution x
2. Or: AT y = 0 has a solution y with yTb ≠ 0.

## Integral equations

$\lambda \phi (x)-\int _{a}^{b}K(x,y)\phi (y)\,dy=0$ and the inhomogeneous equation

$\lambda \phi (x)-\int _{a}^{b}K(x,y)\phi (y)\,dy=f(x).$ The Fredholm alternative states that, for any non-zero fixed complex number $\lambda \in {\mathbb {C} }$ , either the first equation has a non-trivial solution, or the second equation has a solution for all $f(x)$ .

A sufficient condition for this theorem to hold is for $K(x,y)$ to be square integrable on the rectangle $[a,b]\times [a,b]$ (where a and/or b may be minus or plus infinity).

## Functional analysis

Results on the Fredholm operator generalize these results to vector spaces of infinite dimensions, Banach spaces.

### Correspondence

Loosely speaking, the correspondence between the linear algebra version, and the integral equation version, is as follows: Let

$T=\lambda -K$ or, in index notation,

$T(x,y)=\lambda \delta (x-y)-K(x,y)$ $T:V\to V$ is given by

$\phi \mapsto \psi$ $\psi (x)=\int _{a}^{b}T(x,y)\phi (y)\,dy$ In this language, the integral equation alternatives are seen to correspond to the linear algebra alternatives.

### Alternative

In more precise terms, the Fredholm alternative only applies when K is a compact operator. From Fredholm theory, smooth integral kernels are compact operators. The Fredholm alternative may be restated in the following form: a nonzero $\lambda$ is either an eigenvalue of K, or it lies in the domain of the resolvent

$R(\lambda ;K)=(K-\lambda \operatorname {Id} )^{-1}.$ 