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  "body": "\n## Definition\n\nLet $k \\ge 1$ be an integer or $\\infty$.\nA *Hilbert manifold of class $C^k$* is a [[Banach manifold|VB95VVVDM8]] of class $C^k$ with an atlas of charts with values in real [[Hilbert spaces|PYYXEXMARY]].\nIt is *modelled on* the Hilbert space $H$ if all charts can be taken with values in $H$.\nMany authors also require the underlying space to be Hausdorff and second countable, or the model $H$ to be separable.\n\n## Remarks\n\nExample: let $H$ be a real Hilbert space and $S = \\set{x \\in H : \\norm{x} = 1}$ its unit sphere.\nFor $p \\in S$, the map $x \\mapsto x - \\ip{x}{p} p$ restricted to $\\set{x \\in S : \\ip{x}{p} \u003e 0}$ is a chart onto the open unit ball of the closed hyperplane $p^\\perp$, with inverse $y \\mapsto y + \\sqrt{1 - \\norm{y}^2}\\, p$.\nBoth maps are smooth, $p^\\perp$ is a Hilbert space, and these charts cover $S$, so $S$ is a smooth Hilbert manifold.\n\nNon-example: the [[Banach space|0BDTTM24DP]] $\\ell^1$ with its identity chart is a Banach manifold, but no compatible chart takes values in a Hilbert space.\nA compatible chart $\\varphi$ into a Hilbert space $H$ would have a [[derivative|06EH97V2A7]] at a point that is a linear isomorphism of Banach spaces from $\\ell^1$ onto $H$.\nBut $\\ell^1$ is not isomorphic to a Hilbert space.\n\n\nEvery finite-dimensional smooth manifold is a Hilbert manifold modelled on $\\R^n$.\n",
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