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  "body": "\n## Statement\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $X$, $Y$ and $Z$ be [[Banach spaces|06GZ956657]] over $\\mathbb{K}$.\nLet $\\Omega \\subseteq X \\times Y$ be open, and let $F \\colon \\Omega \\to Z$ be of class $C^1$ in the sense of the [[Fréchet derivative|06EH97V2A7]].\nLet $D_x F$ and $D_y F$ be the partial derivatives of $F$, with values in the continuous linear maps $X \\to Z$ and $Y \\to Z$.\nLet $(a, b) \\in \\Omega$ with $F(a, b) = 0$, and suppose that $D_y F(a, b) \\colon Y \\to Z$ is bijective.\nThen there are open sets $U \\subseteq X$ and $V \\subseteq Y$ with $a \\in U$, $b \\in V$ and $U \\times V \\subseteq \\Omega$, and a unique map $g \\colon U \\to V$, such that the following holds.\nFor $(x, y) \\in U \\times V$, $F(x, y) = 0$ if and only if $y = g(x)$.\nAfter shrinking $U$ if needed, $g$ is of class $C^1$, $D_y F(x, g(x))$ is bijective for each $x \\in U$, and\n$$Dg(x) = -D_y F(x, g(x))^{-1}\\, D_x F(x, g(x)).$$\n\n## Remarks\n\nApply the [[inverse function theorem in Banach spaces|M218MWZ62X]] to $\\Phi(x, y) = (x, F(x, y))$ from $\\Omega$ to $X \\times Z$.\nIts derivative at $(a, b)$ is bijective because $D_y F(a, b)$ is, and the local inverse of $\\Phi$ has the form $(x, z) \\mapsto (x, h(x, z))$; then $g(x) = h(x, 0)$.\nThe formula for $Dg$ comes from differentiating $F(x, g(x)) = 0$ with the chain rule.\n\nExample: let $h \\in C([0,1])$ and $F(\\lambda, u) = u - h - \\lambda \\sin \\circ u$ on $\\R \\times C([0,1])$.\nThen $F(0, h) = 0$ and $D_u F(0, h) = \\id$.\nSo for $\\lambda$ near $0$, the equation $u = h + \\lambda \\sin(u)$ has exactly one solution $u = g(\\lambda)$ near $h$, and $g$ is $C^1$.\n\nFor finite-dimensional $X$, $Y$ and $Z$, the theorem is the [[implicit function theorem|0BXBADR94C]] of several-variable calculus.\n\n## Explanations\n\n### L1 human\n\nThe theorem says when an equation $F(x, y) = 0$ can be solved for $y$ as a function of $x$ near a known solution.\nThe test is linear: the derivative in $y$ must be invertible.\nIt works when $x$ and $y$ are functions themselves, as long as the spaces are complete normed spaces.\n\n### L2 human\n\nThe theorem is the basic tool of perturbation theory.\nA solution of a problem at one parameter value persists for nearby parameters when the linearized problem is uniquely solvable.\nThe solution then depends smoothly on the parameter.\nBifurcation theory studies exactly the points where $D_y F$ fails to be invertible.\nIt is equivalent to the [[inverse function theorem in Banach spaces|M218MWZ62X]], which is the special case $F(x, y) = f(y) - x$.\n",
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