Banach analytic space

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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].


[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)
How to Cite This Entry:
Banach analytic space. D.A. Ponomarev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098