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  "body": "\n## Statement\n\nA *graded* [[Fréchet space|3GAA614673]] is a Fréchet space $F$ with a fixed increasing sequence of seminorms $\\norm{\\cdot}_0 \\le \\norm{\\cdot}_1 \\le \\norm{\\cdot}_2 \\le \\cdots$ that defines its topology.\nLet $F$ and $G$ be graded Fréchet spaces and let $U \\subseteq F$ be open.\nA continuous map $P \\colon U \\to G$ is *tame* if each point of $U$ has a neighbourhood $V$ with this property.\nThere are integers $r, b \\ge 0$ and constants $C_n$ with $\\norm{P(f)}_n \\le C_n(1 + \\norm{f}_{n+r})$ for all $f \\in V$ and all $n \\ge b$.\nIt is *smooth tame* if it is smooth and all its derivatives are tame.\nA graded Fréchet space is *tame* if it is a tame direct summand of a space of exponentially decreasing sequences in a Banach space.\nFor a compact manifold $M$, the space $C^\\infty(M)$ graded by the $C^n$ norms is tame.\n\nLet $F$ and $G$ be tame Fréchet spaces, let $U \\subseteq F$ be open, and let $P \\colon U \\to G$ be a smooth tame map.\nSuppose that for every $f \\in U$ the [[derivative|06EH97V2A7]] $DP(f) \\colon F \\to G$ is bijective, and that the family of inverses $VP \\colon U \\times G \\to F$, $VP(f, k) = DP(f)^{-1} k$, is a smooth tame map.\nThen $P$ is locally invertible: each $f_0 \\in U$ has an open neighbourhood $U_0$ such that $P(U_0)$ is open and $P \\colon U_0 \\to P(U_0)$ is a bijection.\nMoreover, each such local inverse is a smooth tame map.\n\n## Remarks\n\nThis is the form of the theorem given by Hamilton; Nash introduced the method to prove his embedding theorem for Riemannian manifolds, and Moser developed it as an abstract iteration.\nThe [[inverse function theorem in Banach spaces|M218MWZ62X]] does not apply, because in typical applications $DP(f)^{-1}$ loses derivatives: it maps $C^{n+r}$ data only to $C^{n}$ solutions.\nThe proof runs a Newton iteration $f_{j+1} = f_j - S_j\\, VP(f_j, P(f_j) - g)$, where $S_j$ are smoothing operators that cut off high frequencies.\nThe quadratic convergence of Newton's method beats the loss of derivatives, and the tame estimates make this precise.\n\nExample: in Kolmogorov–Arnold–Moser theory and in isometric embeddings of Riemannian manifolds, the linearised equations lose derivatives.\nThere the theorem gives solutions of the nonlinear equations near a known one.\n\n## Explanations\n\n### L1 human\n\nThe usual inverse function theorem says that if the derivative of a map is invertible, then the map itself is invertible near the point.\nFor maps between spaces of smooth functions this can fail when the inverse of the derivative makes functions rougher.\nThe Nash–Moser theorem restores the conclusion when every map involved obeys polynomial-type bounds that lose only a fixed number of derivatives.\n\n### L2 human\n\nThe theorem is a tool for nonlinear PDEs whose linearisation is solvable only with a loss of derivatives, such as small divisor problems and isometric embeddings.\nTo apply it, one first checks that the nonlinear map is smooth tame, which is usually routine for differential operators.\nOne then checks that the linearised problem is solvable at every point near the base point, with tame estimates.\nThe second check is the real work, and it is often done with energy estimates or with Fourier analysis.\n",
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