Banach analytic space

Summary

A Banach analytic space is an infinite-dimensional analogue of a complex analytic space. It is modelled locally on the zero set of a holomorphic map from an open subset of a complex Banach space to another one. The notion was introduced by Douady to construct the space of compact complex subspaces of a given complex space. Complex analytic spaces and Banach manifolds over $\C$ are Banach analytic spaces. They are used in deformation theory. Spaces of maps or of subspaces are first built as Banach analytic spaces, and then shown to be finite-dimensional.