Fredholm operators

Fredholm operators are a class of linear transformations between infinite-dimensional spaces that behave similarly to matrices with finite-dimensional deviation. They are characterized by having a finite-dimensional kernel (solutions mapped to zero), a finite-dimensional cokernel, and a closed range. These properties allow for a well-defined "index," which is the difference between the dimensions of the kernel and cokernel, playing a crucial role in analyzing solutions to operator equations. Fredholm operators are essential in functional analysis, spectral theory, and partial differential equations, providing a robust framework for understanding stability, solvability, and perturbation behavior in infinite-dimensional contexts.