Fredholm Alternative

For any nonzero , either

1. The equation has a nonzero solution , or

2. The equation has a unique solution for any function .

In the second case, the solution depends continuously on . The Fredholm alternative applies when is a compact operator, such as an integral operator with a smooth integral kernel.

The Fredholm alternative can be restated as follows: any nonzero which is not an eigenvalue of a compact operator is in the resolvent, i.e., , is bounded. The basic special case is when is finite-dimensional, in which case any nondegenerate matrix is diagonalizable.