Fredholm Theory

Fredholm theory deals with certain types of equations involving operators—mathematical tools that map functions to other functions—especially in infinite-dimensional spaces. It provides criteria for when these equations have solutions, focusing on operators called Fredholm operators. These operators have well-behaved properties, such as finite-dimensional kernels (solutions to the homogeneous equation) and cokernels (constraints on solutions). Fredholm theory helps analyze whether solutions exist, are unique, or depend continuously on the input, and it is fundamental in solving integral equations and understanding stability in various mathematical and physical systems.