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  "body": "\n## Definition\n\nLet $F$ and $G$ be real [[Fréchet spaces|3GAA614673]], and let $U \\subseteq F$ be open.\nA map $f \\colon U \\to G$ is of class $C^1$ if it is continuous, the [[Gateaux derivative|EAZF9NJW12]] $Df(x) h = \\lim_{t \\to 0} (f(x + t h) - f(x)) / t$ exists for all $x \\in U$ and $h \\in F$, and $Df \\colon U \\times F \\to G$ is continuous.\nIt is of class $C^{k+1}$ if it is $C^1$ and $Df$ is $C^k$, and *smooth* if it is $C^k$ for all $k$.\n\nLet $M$ be a [[Hausdorff space|0MXB4CKHFW]].\nA *smooth Fréchet atlas* on $M$ is a family of charts $(U_i, \\varphi_i)_{i \\in I}$ such that:\n\n1. Each $U_i$ is open in $M$ and the sets $U_i$ cover $M$.\n2. Each $\\varphi_i$ is a [[homeomorphism|0QT00PB5G2]] from $U_i$ onto an [[open subset|02BH3MCBQB]] of a Fréchet space.\n3. Each transition map $\\varphi_j \\circ \\varphi_i^{-1} \\colon \\varphi_i(U_i \\cap U_j) \\to \\varphi_j(U_i \\cap U_j)$ is smooth.\n\nA *Fréchet manifold* is a space $M$ with an equivalence class of smooth Fréchet atlases, two atlases being equivalent when their union is an atlas.\n\n## Remarks\n\nEvery Banach space is a Fréchet space, and every smooth [[Banach manifold|9K15AMGBV9]] is a Fréchet manifold.\n\nExample: the Fréchet space $C^\\infty(S^1)$ of smooth functions on the circle, with the single chart $\\id$, is a Fréchet manifold.\nMore generally, for a compact manifold $K$ and a manifold $N$, the space $C^\\infty(K, N)$ of smooth maps is a Fréchet manifold.\nThe diffeomorphisms of $K$ form a Fréchet Lie group.\n\nNon-example: take charts with values in the space of compactly supported smooth functions on $\\R$ with the sup norm.\nThat normed space is not complete, so it is not a Fréchet space and clause 2 fails.\n",
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