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  "body": "\n## Summary\n\nA Fredholm map is a differentiable map between [[Banach manifolds|9K15AMGBV9]] whose derivative at every point is a [[Fredholm operator|8HWGKWD6ZM]].\nIts index, the index of the derivative, is locally constant, and it plays the role of the difference of dimensions of source and target.\nNonlinear elliptic equations on compact manifolds give Fredholm maps between Sobolev spaces of maps.\nThe [[Sard–Smale theorem|Z0NPFYDVP2]] extends Sard's theorem to Fredholm maps, so that the solution spaces of generic perturbed elliptic equations are finite-dimensional manifolds.\n\n## Explanations\n\n### L1 human\n\nA map between finite-dimensional spaces of dimensions $m$ and $n$ has, at a regular point, solution sets of dimension $m - n$.\nA Fredholm map is a map between infinite-dimensional spaces that still behaves like this: infinitely many equations in infinitely many unknowns, but with a finite and well-defined difference.\nThat difference is the index.\nFor instance, the map $(x_1, x_2, x_3, \\dots) \\mapsto (x_2, x_3, \\dots)$ on $\\ell^2$ is linear and Fredholm of index $1$: one unknown is left free.\n",
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