# 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 is a linear transformation, then exactly one of the following holds:

- For each vector
*v*in*V*there is a vector*u*in*V*so that . In other words: T is surjective (and so also bijective, since*V*is finite-dimensional). - .

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:

*Either:**A***x**=**b**has a solution**x***Or:**A*^{T}**y**= 0 has a solution**y**with**y**^{T}**b**≠ 0.

In other words, *A* **x** = **b** has a solution if and only if for any **y** s.t. *A*^{T} **y** = 0, **y**^{T}**b** = 0 .

## Integral equations

Let be an integral kernel, and consider the homogeneous equation, the Fredholm integral equation,

and the inhomogeneous equation

The Fredholm alternative states that, for any non-zero fixed complex number , either the first equation has a non-trivial solution, or the second equation has a solution for all .

A sufficient condition for this theorem to hold is for to be square integrable on the rectangle (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

or, in index notation,

with the Dirac delta function. Here, *T* can be seen to be an linear operator acting on a Banach space *V* of functions , so that

is given by

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 is either an eigenvalue of *K*, or it lies in the domain of the resolvent

## See also

## References

- E.I. Fredholm, "Sur une classe d'equations fonctionnelles",
*Acta Math.*,**27**(1903) pp. 365–390. - A. G. Ramm, "A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators",
*American Mathematical Monthly*,**108**(2001) p. 855. - {{#invoke:citation/CS1|citation

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