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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b0151401.png" /> of an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b0151402.png" /> in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b0151403.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b0151404.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b0151405.png" /> is an analytic mapping into the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b0151406.png" />. As distinct from the finite-dimensional case, not one structure sheaf, but a set of sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b0151407.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b0151408.png" /> is an open set in an arbitrary Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b0151409.png" />, is defined on the local model. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514010.png" /> is defined as the quotient of the sheaf of germs of analytic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514011.png" /> by the subsheaf of germs of mappings of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514013.png" /> is a local analytic mapping, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514014.png" /> is generated by mappings which assume values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514015.png" />. The sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514016.png" /> define a functor from the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514017.png" /> of open sets in Banach spaces and their analytic mappings into the category of sheaves of sets on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514018.png" />.

A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514019.png" /> with a functor from the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514020.png" /> into the category of sheaves of sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514021.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514022.png" /> has a neighbourhood that is isomorphic to a model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514023.png" /> and for which there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514024.png" /> inducing an automorphism of the model and having a completely-continuous differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514025.png" /> [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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514026.png" /> that are closed and admit closed complements.

Finitely-defined Banach analytic sets, i.e. models of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015140/b01514027.png" />, have local properties which correspond to classical properties: primary decomposition, Hilbert's Nullstellen theorem, the local description theorem, etc., are all applicable [2].

References

[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)