A Fredholm operator is a type of linear operator that arises in functional analysis, particularly in the study of Hilbert spaces and Banach spaces. It is characterized by having a finite-dimensional kernel (the set of solutions to the equation Ax = 0) and a closed range. This means that the operator can be inverted on its range, leading to important implications in solving differential equations and other mathematical problems. Fredholm operators are classified into three types based on their index, which is the difference between the dimension of the kernel and the dimension of the cokernel (the quotient space of the range). The index helps determine the solvability of equations involving these operators, making them essential in various applications, including quantum mechanics and control theory.