Fredholm Alternative

The Fredholm alternative for a compact operator on a complex Banach space states that for any nonzero , exactly one of the following holds:

1. The equation has a nonzero solution , or

2. The equation has a unique solution for every element .

In the second case, the solution depends continuously on . The Fredholm alternative applies when is a compact operator, such as an integral operator on the Banach space of continuous functions on a closed interval , with a continuous integral kernel on .

The Fredholm alternative can be restated as follows: any nonzero which is not an eigenvalue of a compact operator belongs to its resolvent set, i.e., exists and is bounded. The basic special case is when is finite-dimensional, in which case any nonsingular matrix has a matrix inverse; diagonalizability is not required.