Fredholm theory is a branch of functional analysis that deals with Fredholm operators, which are a special class of linear operators. These operators are defined on a Banach space and have a finite-dimensional kernel and cokernel. The theory provides important insights into the solvability of linear equations and the structure of the solution space. A key result in Fredholm theory is the Fredholm alternative, which states that for a compact operator, either the homogeneous equation has only the trivial solution, or the inhomogeneous equation has a solution for every right-hand side. This framework is essential in various applications, including partial differential equations and quantum mechanics.