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  "body": "\n## Summary\n\nA Banach–Lie group is a group that is also a smooth [[Banach manifold|9K15AMGBV9]] in which multiplication and inversion are smooth.\nIt generalises a finite-dimensional [[Lie group|7JBRGCVJRB]] to infinite dimension.\nIts Lie algebra is a Banach–Lie algebra, and, unlike for more general infinite-dimensional groups, the exponential map is still a local diffeomorphism at $0$.\nBasic examples are the group of invertible bounded operators on a Banach space and the unitary group of a Hilbert space with the norm topology.\n\n\n## Explanations\n\n### L1 human\n\nA Lie group is a group whose elements form a smooth space, such as the rotations of space.\nA Banach–Lie group is the same idea when the group has infinitely many directions.\nFor example, the invertible bounded operators on the Hilbert space $\\ell^2$ form a group, and near the identity they look like a ball of the space of all bounded operators.\nMany tools of finite-dimensional Lie theory, such as the exponential map, keep working in this setting.\n",
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