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  "body": "\n## Definition\n\nLet $M$ be a [[Hausdorff space|0MXB4CKHFW]] and let $k \\ge 1$ be an integer or $\\infty$.\nA *chart* on $M$ is a pair $(U, \\varphi)$ with $U \\subseteq M$ open.\nHere $\\varphi$ is a [[homeomorphism|0QT00PB5G2]] from $U$ onto an [[open subset|02BH3MCBQB]] of a real [[Banach space|06GZ956657]] $E$.\nA *$C^k$ atlas* on $M$ is a family of charts $(U_i, \\varphi_i)_{i \\in I}$ such that:\n\n1. The sets $U_i$ cover $M$.\n2. For all $i, j \\in I$, the 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 of class $C^k$ in the sense of the [[Fréchet derivative|06EH97V2A7]].\n\nTwo $C^k$ atlases are equivalent if their union is a $C^k$ atlas.\nA *Banach manifold of class $C^k$* is a space $M$ with an equivalence class of $C^k$ atlases.\nIf all charts take values in the same Banach space $E$, the manifold is *modelled on* $E$.\n\n## Remarks\n\nWhen every model space is $\\R^n$, this is the definition of a [[smooth manifold|F45VQ47GPV]] of class $C^k$.\nWhen every model space is a Hilbert space, $M$ is a Hilbert manifold.\nSome authors start from a set instead of a Hausdorff space and give it the topology defined by the charts.\n\nExample: an open subset $U$ of a Banach space $E$, with the single chart $(U, \\id)$, is a Banach manifold.\nSo the invertible elements of the Banach algebra of bounded operators on $E$ form a Banach manifold, since they form an open subset.\nThe unit sphere of a Hilbert space is a Banach manifold, with charts given by projections onto tangent hyperplanes.\n\nNon-example: on $\\R$, the charts $(\\R, \\id)$ and $(\\R, x \\mapsto x^3)$ do not form a $C^1$ atlas together.\nThe transition map $x \\mapsto x^{1/3}$ is not differentiable at $0$, so clause 2 fails.\n",
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