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  "body": "\n## Definition\n\nA *graded Fréchet space* is a [[Fréchet space|N8TGDDJCFJ]] $F$ with a sequence of [[seminorms|4KX81Q85DT]] $\\norm{\\cdot}_0 \\le \\norm{\\cdot}_1 \\le \\norm{\\cdot}_2 \\le \\cdots$ that defines its topology.\nLet $L \\colon F \\to G$ be a [[continuous linear map|0S3SMRH0Z5]] between graded Fréchet spaces.\nIt is *tame* if there are integers $r, b \\ge 0$ and constants $C_n$ with $\\norm{L f}_n \\le C_n \\norm{f}_{n + r}$ for all $f \\in F$ and all $n \\ge b$.\n\nFor a [[Banach space|0BDTTM24DP]] $B$, let $\\Sigma(B)$ be the space of sequences $f = (f_k)_{k \\ge 0}$ in $B$ with\n$$\\norm{f}_n = \\sum_{k \\ge 0} e^{n k} \\norm{f_k}_B \u003c \\infty \\quad \\text{for all } n \\ge 0,$$\ngraded by these norms.\nA graded Fréchet space $F$ is *tame* if there are a Banach space $B$ and tame linear maps $L \\colon F \\to \\Sigma(B)$ and $M \\colon \\Sigma(B) \\to F$ with $M \\circ L = \\id_F$.\n\n## Remarks\n\nExample: $\\Sigma(B)$ itself is tame, with $L = M = \\id$; the identity satisfies the estimate with $r = b = 0$ and $C_n = 1$.\nHamilton proved that $C^\\infty(X)$ for a compact manifold $X$, graded by the $C^n$ norms, is tame.\n\nNon-example of a tame map: on $\\Sigma(\\R)$ let $(L f)_k = f_{\\floor{k/2}}$.\nThen $\\norm{L f}_n \\le (1 + e^n) \\norm{f}_{2n}$, so $L$ is continuous.\nFor the unit sequence $\\delta_j$ one gets $\\norm{L \\delta_j}_n \\ge e^{2nj}$ while $\\norm{\\delta_j}_{n+r} = e^{(n+r)j}$, so for $n \u003e r$ no constant $C_n$ works for all $j$, and $L$ is not tame.\n\nThe grading is part of the data: equivalent gradings, with $\\norm{f}_n$ and $\\norm{f}'_n$ bounded by each other up to a fixed shift $r$ and constants, give the same tame maps.\n",
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