A Fredholm operator is a type of linear operator that arises in functional analysis, particularly in the study of integral equations and differential equations. It is defined on a Banach space and has a finite-dimensional kernel (the set of solutions to the equation Ax = 0) and a finite-dimensional cokernel (the quotient of the codomain by the image of the operator). This structure allows for the analysis of the operator's properties, such as invertibility. Fredholm operators are classified into three types based on their index, which is the difference between the dimensions of the kernel and cokernel. The index provides important information about the solvability of associated equations. In applications, Fredholm operators are crucial in areas like quantum mechanics, control theory, and mathematical physics, where they help in understanding the behavior of systems described by linear equations.