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  "body": "\n## Definition\n\nA *Banach–Lie group* is a [[group|G5S1M9]] $G$ with the structure of a smooth [[Banach manifold|VB95VVVDM8]] such that:\n\n1. The multiplication $G \\times G \\to G$, $(g, h) \\mapsto g h$, is smooth.\n2. The inversion $G \\to G$, $g \\mapsto g^{-1}$, is smooth.\n\n## Remarks\n\nExample: let $E$ be a real [[Banach space|0BDTTM24DP]] and $\\mathcal{L}(E)$ the Banach space of bounded linear operators with the operator norm.\nThe group $GL(E)$ of invertible operators is an open subset of $\\mathcal{L}(E)$, so it is a Banach manifold with one chart.\nMultiplication is the restriction of a continuous bilinear map, hence smooth, and inversion is smooth because near $A$ it is given by the convergent series $(A - X)^{-1} = \\sum_{k \\ge 0} (A^{-1} X)^k A^{-1}$.\n\nNon-example: on $\\R$ with its standard smooth structure, define $x * y = (x^{1/3} + y^{1/3})^3$.\nThis is a group, isomorphic to $(\\R, +)$ through $x \\mapsto x^{1/3}$, but the partial derivative of $x * y$ in $x$ at the point $(0, 1)$ does not exist, so clause 1 fails.\n\nThe tangent space at the identity carries a Lie bracket and is a Banach–Lie algebra.\nEvery finite-dimensional Lie group is a Banach–Lie group.\n",
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