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  "body": "\n## Statement\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\\mathbb{K}$.\nLet $U \\subseteq X$ be open, and let $f \\colon U \\to Y$ be of class $C^1$.\nThis means that $f$ has a [[Fréchet derivative|06EH97V2A7]] $Df(x)$ at each $x \\in U$ and that $x \\mapsto Df(x)$ is continuous for the [[operator norm|06EKW1AKXF]].\nLet $a \\in U$, and suppose that $Df(a) \\colon X \\to Y$ is bijective.\nThen there are open sets $V \\subseteq U$ with $a \\in V$ and $W \\subseteq Y$ with $f(a) \\in W$ such that $f$ maps $V$ bijectively onto $W$.\nThe inverse $g \\colon W \\to V$ is of class $C^1$, and $Dg(f(x)) = Df(x)^{-1}$ for each $x \\in V$.\nIf $f$ is of class $C^k$ with $k \\ge 1$, then so is $g$.\n\n## Remarks\n\nBy the [[bounded inverse theorem|MR5FJXR348]], the inverse of the bijective continuous linear map $Df(a)$ is continuous.\nTo solve $f(x) = y$ for $y$ near $f(a)$, one looks for a fixed point of $x \\mapsto x - Df(a)^{-1}(f(x) - y)$.\nNear $a$ this map is a contraction, because $Df(x)$ is close to $Df(a)$; the [[Banach fixed point theorem|0VK6VEH345]] gives existence and uniqueness.\nSmoothness of the inverse follows from the formula for $Dg$ and the continuity of inversion of operators.\n\nExample: on $X = Y = C([0,1])$ with the sup norm, let $f(u) = u + u^3$, computed pointwise.\nThen $Df(0) = \\id$, so each $v$ close to $0$ equals $u + u^3$ for exactly one $u$ close to $0$, and $u$ depends smoothly on $v$.\n\nThe hypothesis $C^1$ cannot be weakened to differentiability at each point.\nOn $\\R$, the map $f(x) = x + 2x^2 \\sin(1/x)$, $f(0) = 0$, has $f'(0) = 1$ but is not injective on any neighbourhood of $0$.\nFor $X = Y = \\R^n$, the theorem is the [[inverse function theorem|0K9HSJFCMJ]] of several-variable calculus.\n\n## Explanations\n\n### L1 human\n\nIf the linear approximation of a smooth map at a point is invertible, the map itself is invertible near that point.\nThis holds even when the spaces are spaces of functions, as long as they are complete normed spaces.\nSo a nonlinear equation $f(x) = y$ can be solved for $y$ near $f(a)$ when its linearization at $a$ can be solved.\n\n### L2 human\n\nThe theorem is the main tool of local nonlinear analysis.\nIt gives the [[implicit function theorem in Banach spaces|BTM6WJ0A52]], the local structure of Banach manifolds and the local existence for many nonlinear differential equations.\nIts proof needs completeness, through the contraction principle.\nIn Fréchet spaces, such as $C^\\infty$ function spaces, it fails in general, and the Nash–Moser theorem replaces it under extra hypotheses.\n",
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