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  "body": "\n## Definition\n\nLet $M$ and $N$ be [[Banach manifolds|VB95VVVDM8]] of class $C^k$ with $k \\ge 1$, and let $f \\colon M \\to N$ be a map of class $C^1$.\nThe map $f$ is a *Fredholm map* if for every $x \\in M$ the [[derivative|06EH97V2A7]] $d f_x \\colon T_x M \\to T_{f(x)} N$ is a [[Fredholm operator|Q5B5YGY7D6]].\nThis means that the kernel of $d f_x$ is finite-dimensional and its image is closed and of finite codimension.\nThe *index* of $f$ at $x$ is the [[Fredholm index|ADJZ99KY2Q]]\n$$\\operatorname{ind}_x f = \\dim \\ker d f_x - \\dim \\operatorname{coker} d f_x.$$\n\n## Remarks\n\nThe Fredholm operators form an open set on which the index is locally constant, and $x \\mapsto d f_x$ is continuous in local charts.\nSo the index is locally constant in $x$, and on a connected $M$ it is a single integer.\n\n\nExample: every $C^1$ map $f \\colon \\R^m \\to \\R^n$ is Fredholm of index $m - n$, since $\\dim \\ker A - \\dim \\operatorname{coker} A = m - n$ for every linear map $A \\colon \\R^m \\to \\R^n$ by rank-nullity.\n\nNon-example: the inclusion $\\iota \\colon \\ell^2 \\to \\ell^2 \\times \\ell^2$, $x \\mapsto (x, 0)$, is smooth and its derivative is injective with closed image, but the cokernel is isomorphic to $\\ell^2$, which is infinite-dimensional, so $\\iota$ is not Fredholm.\n\nA map of the form $f = L + K$ with $L$ a linear Fredholm operator and $K$ a $C^1$ map whose derivatives are compact operators is Fredholm with the index of $L$.\n",
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