Fredholm theorems

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for integral equations

Theorem 1.

The homogeneous equation


and its transposed equation


have, for a fixed value of the parameter , either only the trivial solution, or have the same finite number of linearly independent solutions: ; .

Theorem 2.

For a solution of the inhomogeneous equation


to exist it is necessary and sufficient that its right-hand side be orthogonal to a complete system of linearly independent solutions of the corresponding homogeneous transposed equation (2):


Theorem 3.

(the Fredholm alternative). Either the inhomogeneous equation (3) has a solution, whatever its right-hand side , or the corresponding homogeneous equation (1) has non-trivial solutions.

Theorem 4.

The set of characteristic numbers of equation (1) is at most countable, with a single possible limit point at infinity.

For the Fredholm theorems to hold in the function space it is sufficient that the kernel of equation (3) be square-integrable on the set ( and may be infinite). When this condition is violated, (3) may turn out to be a non-Fredholm integral equation. When the parameter and the functions involved in (3) take complex values, then instead of the transposed equation (2) one often considers the adjoint equation to (1):

In this case condition (4) is replaced by

These theorems were proved by E.I. Fredholm [1].


[1] E.I. Fredholm, "Sur une classe d'equations fonctionnelles" Acta Math. , 27 (1903) pp. 365–390


Instead of the phrases "transposed equation" and "adjoint equation" one sometimes uses "adjoint equation of a Fredholm integral equationadjoint equation" and "conjugate equation of a Fredholm integral equationconjugate equation" (cf. [a4]); in the latter terminology is replaced by .


[a1] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
[a2] K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970)
[a3] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
[a4] P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)
How to Cite This Entry:
Fredholm theorems. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article