Banach analytic space

Definition

A model is a triple $(U, F, f)$ where $E$ and $F$ are complex Banach spaces, $U \subseteq E$ is open and $f \colon U \to F$ is a holomorphic map. Its underlying space is $Z = f^{-1}(0)$ with the topology induced from $U$. For each complex Banach space $G$, the model defines a sheaf $\mathcal{H}_Z(G)$ on $Z$. Its stalk at $z \in Z$ is a quotient of the space of germs at $z$ of holomorphic maps $g \colon U \to G$. Let $L(F, G)$ be the Banach space of continuous linear maps $F \to G$. One divides by the germs $x \mapsto \varphi(x)\bigl(f(x)\bigr)$, where $\varphi$ is a germ at $z$ of a holomorphic map $U \to L(F, G)$. These sheaves are the holomorphic maps from $Z$ to $G$.

A Banach analytic space is a Hausdorff space $X$ with the following data and property. For each complex Banach space $G$ there is a sheaf $\mathcal{H}_X(G)$ of complex vector spaces on $X$, depending functorially on $G$. Its sections play the role of holomorphic maps to $G$. Each point of $X$ has an open neighbourhood isomorphic to the underlying space of a model, compatibly with the sheaves $\mathcal{H}(G)$ for all $G$.