Fredholm's theorem

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: :(\operatorname{row } M)^\bot = \ker M. Similarly, the orthogonal complement of the column space of M is the null space of the adjoint: :(\operatorname{col } M)^\bot = \ker M^*. ==Integral equations==

Fredholm's theorem for integral equations is expressed as follows. Let K(x,y) be an integral kernel, and consider the homogeneous equations :\int_a^b K(x,y) \phi(y) \,dy = \lambda \phi(x) and its complex adjoint :\int_a^b \psi(x) \overline{K(x,y)} \, dx = \overline {\lambda}\psi(y). Here, \overline{\lambda} denotes the complex conjugate of the complex number \lambda, and similarly for \overline{K(x,y)}. Then, Fredholm's theorem is that, for any fixed value of \lambda, these equations have either the trivial solution \psi(x)=\phi(x)=0 or have the same number of linearly independent solutions \phi_1(x),\cdots,\phi_n(x), \psi_1(y),\cdots,\psi_n(y). 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). 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. ==Existence of solutions==

One of Fredholm's theorems, closely related to the Fredholm alternative, concerns the existence of solutions to the inhomogeneous Fredholm equation : \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 f(x) is orthogonal to the complete set of solutions \{\psi_n(x)\} of the corresponding homogeneous adjoint equation: :\int_a^b \overline{\psi_n(x)} f(x) \,dx=0 where \overline{\psi_n(x)} is the complex conjugate of \psi_n(x) and the former is one of the complete set of solutions to :\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 K(x,y) to be square integrable on the rectangle [a,b]\times[a,b]. ==References==