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  "body": "\n## Definition\n\nA *model* is a triple $(U, F, f)$ where $E$ and $F$ are complex [[Banach spaces|06GZ956657]], $U \\subseteq E$ is open and $f \\colon U \\to F$ is a [[holomorphic map|E8N3EC0YH5]].\nIts underlying space is $Z = f^{-1}(0)$ with the topology induced from $U$.\nFor each complex Banach space $G$, the model defines a [[sheaf|TB6TM6R4MD]] $\\mathcal{H}_Z(G)$ on $Z$.\nIts stalk at $z \\in Z$ is a quotient of the space of germs at $z$ of holomorphic maps $g \\colon U \\to G$.\nLet $L(F, G)$ be the Banach space of continuous linear maps $F \\to G$.\nOne divides by the germs $x \\mapsto \\varphi(x)\\bigl(f(x)\\bigr)$, where $\\varphi$ is a germ at $z$ of a holomorphic map $U \\to L(F, G)$.\nThese sheaves are the holomorphic maps from $Z$ to $G$.\n\nA *Banach analytic space* is a [[Hausdorff space|0S1FJ8VEX9]] $X$ with the following data and property.\nFor each complex Banach space $G$ there is a sheaf $\\mathcal{H}_X(G)$ of complex vector spaces on $X$, depending functorially on $G$.\nIts sections play the role of holomorphic maps to $G$.\nEach point of $X$ has an open neighbourhood isomorphic to the underlying space of a model, compatibly with the sheaves $\\mathcal{H}(G)$ for all $G$.\n\n## Remarks\n\nExample: if $E = \\C^n$ and $F = \\C^k$, then a model is a local model of a [[complex analytic space|HXSZA44J5J]], and the sheaf $\\mathcal{H}_Z(\\C)$ is its structure sheaf.\nIn this case $\\mathcal{H}_Z(G)$ is determined by $\\mathcal{H}_Z(\\C)$, so complex analytic spaces are Banach analytic spaces.\nAlso every open subset $U$ of a complex Banach space is a Banach analytic space, given by the model $(U, \\set{0}, 0)$.\n\nNon-example: the triple $(\\C, \\C, f)$ with $f(z) = \\bar{z}$ is not a model, because $f$ is not holomorphic.\n\nThe definition uses all Banach spaces $G$, not only $G = \\C$.\nWhen $F$ is infinite-dimensional, the map $f$ is not given by finitely many scalar equations, and scalar functions alone do not record the structure that $f$ defines.\nThe notion is due to Douady.\n",
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