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  "body": "\n## Summary\n\nA Banach analytic space is an infinite-dimensional analogue of a [[complex analytic space|HXSZA44J5J]].\nIt is modelled locally on the zero set of a [[holomorphic map|E8N3EC0YH5]] from an open subset of a complex [[Banach space|06GZ956657]] to another one.\nThe notion was introduced by Douady to construct the space of compact complex subspaces of a given complex space.\nComplex analytic spaces and [[Banach manifolds|9K15AMGBV9]] over $\\C$ are Banach analytic spaces.\nThey are used in deformation theory.\nSpaces of maps or of subspaces are first built as Banach analytic spaces, and then shown to be finite-dimensional.\n\n## Explanations\n\n### L1 human\n\nA finite-dimensional complex space is cut out locally by finitely many holomorphic equations in $\\C^n$.\nIn a Banach analytic space the ambient space $\\C^n$ is replaced by an infinite-dimensional Banach space, and the equations take values in another Banach space.\nFor example, any open subset of a complex Banach space is a Banach analytic space with no equations at all.\nThe equation $\\ip{x}{x} = 1$ of the unit sphere in a complex Hilbert space $H$ is not allowed, because $x \\mapsto \\ip{x}{x}$ is not holomorphic.\nBut the zero set of any holomorphic map from $H$ to another Banach space is.\n",
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