Fredholm map

Summary

A Fredholm map is a differentiable map between Banach manifolds whose derivative at every point is a Fredholm operator. Its index, the index of the derivative, is locally constant, and it plays the role of the difference of dimensions of source and target. Nonlinear elliptic equations on compact manifolds give Fredholm maps between Sobolev spaces of maps. The Sard–Smale theorem extends Sard's theorem to Fredholm maps, so that the solution spaces of generic perturbed elliptic equations are finite-dimensional manifolds.