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

==Linear algebra==
If ''V'' is an ''n''-dimensional [[vector space]] and <math>T:V\to V</math> 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 <math>T(u) = v</math>. In other words: T is surjective (and so also bijective, since ''V'' is finite-dimensional).
#<math>\dim(\ker(T)) > 0</math>.

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

#''Either:'' ''A'' '''x''' = '''b''' has a solution '''x'''
#''Or:'' ''A''<sup>T</sup> '''y''' = 0 has a solution '''y''' with '''y'''<sup>T</sup>'''b''' ≠ 0.

In other words, ''A'' '''x''' = '''b''' has a solution <math> (\mathbf{b} \in \operatorname{Im}(A)) </math> if and only if for any '''y''' s.t. ''A''<sup>T</sup> '''y''' = 0, '''y'''<sup>T</sup>'''b''' = 0 <math> (\mathbf{b} \in \ker(A^T)^{\bot}) </math>.
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==Integral equations==
Let <math>K(x,y)</math> be an [[integral kernel]], and consider the [[Homogeneous polynomial|homogeneous equation]], the [[Fredholm integral equation]], 

:<math>\lambda \phi(x)- \int_a^b K(x,y) \phi(y) \,dy = 0</math>

and the inhomogeneous equation

:<math>\lambda \phi(x) - \int_a^b K(x,y) \phi(y) \,dy = f(x).</math>

The Fredholm alternative states that, for any non-zero fixed [[complex number]] <math>\lambda \in \mathbb{C}</math>, either the first equation has a non-trivial solution, or the second equation has a solution for all <math>f(x)</math>.

A sufficient condition for this theorem to hold is for <math>K(x,y)</math> to be [[square integrable]] on the rectangle <math>[a,b]\times[a,b]</math> (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 space]]s.

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

:<math>T=\lambda - K</math>

or, in index notation,

:<math>T(x,y)=\lambda \delta(x-y) - K(x,y)</math>

with <math>\delta(x-y)</math> the [[Dirac delta function]].  Here, ''T'' can be seen to be an [[linear operator]] acting on a Banach space ''V'' of functions <math>\phi(x)</math>, so that

:<math>T:V\to V</math>

is given by

:<math>\phi \mapsto \psi</math>

with <math>\psi</math> given by

:<math>\psi(x)=\int_a^b T(x,y) \phi(y) \,dy</math>

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 <math>\lambda</math> is either an [[eigenvalue]] of ''K'', or it lies in the domain of the [[resolvent formalism|resolvent]]

:<math>R(\lambda; K)= (K-\lambda \operatorname{Id})^{-1}.</math>

== See also ==

*[[Spectral theory of compact operators]]

==References==
* E.I. Fredholm, "Sur une classe d'equations fonctionnelles", ''Acta Math.'', '''27'''  (1903)  pp.&nbsp;365–390.
* A. G. Ramm, "[http://arxiv.org/abs/math/0011133 A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators]", ''American Mathematical Monthly'', '''108''' (2001) p.&nbsp;855.
* {{springer|first=B.V. |last=Khvedelidze|id=f/f041470|title=Fredholm theorems for integral equations}}
* {{mathworld|urlname=FredholmAlternative|title=Fredholm Alternative}}

[[Category:Fredholm theory]]
[[Category:Linear algebra]]