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:

${\displaystyle (\operatorname {row} M)^{\bot }=\ker M.}$ 

Similarly, the orthogonal complement of the column space of M is the null space of the adjoint:

${\displaystyle (\operatorname {col} M)^{\bot }=\ker M^{*}.}$ 

Fredholm's theorem for integral equations is expressed as follows. Let ${\displaystyle K(x,y)}$  be an integral kernel, and consider the homogeneous equations

${\displaystyle \int _{a}^{b}K(x,y)\phi (y)\,dy=\lambda \phi (x)}$ 

and its complex adjoint

${\displaystyle \int _{a}^{b}\psi (x){\overline {K(x,y)}}\,dx={\overline {\lambda }}\psi (y).}$ 

Here, ${\displaystyle {\overline {\lambda }}}$  denotes the complex conjugate of the complex number ${\displaystyle \lambda }$ , and similarly for ${\displaystyle {\overline {K(x,y)}}}$ . Then, Fredholm's theorem is that, for any fixed value of ${\displaystyle \lambda }$ , these equations have either the trivial solution ${\displaystyle \psi (x)=\phi (x)=0}$  or have the same number of linearly independent solutions ${\displaystyle \phi _{1}(x),\cdots ,\phi _{n}(x)}$ , ${\displaystyle \psi _{1}(y),\cdots ,\psi _{n}(y)}$ .

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

Here, the integral is expressed as a one-dimensional integral on the real number line. 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

${\displaystyle \lambda \phi (x)-\int _{a}^{b}K(x,y)\phi (y)\,dy=f(x).}$ 

Solutions to this equation exist if and only if the function ${\displaystyle f(x)}$  is orthogonal to the complete set of solutions ${\displaystyle \{\psi _{n}(x)\}}$  of the corresponding homogeneous adjoint equation:

${\displaystyle \int _{a}^{b}{\overline {\psi _{n}(x)}}f(x)\,dx=0}$ 

where ${\displaystyle {\overline {\psi _{n}(x)}}}$  is the complex conjugate of ${\displaystyle \psi _{n}(x)}$  and the former is one of the complete set of solutions to

${\displaystyle \lambda {\overline {\psi (y)}}-\int _{a}^{b}{\overline {\psi (x)}}K(x,y)\,dx=0.}$ 

A sufficient condition for this theorem to hold is for ${\displaystyle K(x,y)}$  to be square integrable on the rectangle ${\displaystyle [a,b]\times [a,b]}$ .

• E.I. Fredholm, "Sur une classe d'equations fonctionnelles", Acta Math., 27 (1903) pp. 365–390.
• Weisstein, Eric W. "Fredholm's Theorem". MathWorld.
• B.V. Khvedelidze (2001) [1994], "Fredholm theorems", Encyclopedia of Mathematics, EMS Press