Banach analytic space

An infinite-dimensional generalization of the concept of an analytic space, which arose in the context of the study of deformations of analytic structures (cf. Deformation). Here, the local model is a Banach analytic set, i.e. a subset  of an open set  in a Banach space over , where  is an analytic mapping into the Banach space . As distinct from the finite-dimensional case, not one structure sheaf, but a set of sheaves , where  is an open set in an arbitrary Banach space , is defined on the local model.  is defined as the quotient of the sheaf of germs of analytic mappings  by the subsheaf of germs of mappings of the type , where  is a local analytic mapping, while  is generated by mappings which assume values in . The sheaves  define a functor from the category  of open sets in Banach spaces and their analytic mappings into the category of sheaves of sets on .

A topological space  with a functor from the category  into the category of sheaves of sets in  in which all points have neighbourhoods isomorphic to some local model, is said to be a Banach analytic space.

Complex-analytic spaces form a complete subcategory in the category of Banach analytic spaces. A Banach analytic space is finite-dimensional if each one of its points  has a neighbourhood that is isomorphic to a model  and for which there exists a mapping  inducing an automorphism of the model and having a completely-continuous differential  [1].

A second special case of a Banach analytic space is a Banach analytic manifold, i.e. an analytic space that is locally isomorphic to open subsets of Banach spaces. An important example is the manifold of linear subspaces of a Banach space over  that are closed and admit closed complements.

Finitely-defined Banach analytic sets, i.e. models of the type , have local properties which correspond to classical properties: primary decomposition, Hilbert's Nullstellen theorem, the local description theorem, etc., are all applicable [2].

#### References[edit]

[1] |  A. Douady, "Les problèmes des modules pour les sous-espace analytique compacts d'une espace analytique donné" Ann. Inst. Fourier (Grenoble) , 16 : 1 (1966) pp. 1–95

|

[2] |  J.-P. Ramis, "Sous-ensembles analytiques d'une variété banachique complexe" , Springer (1970)