An analytic space over the field of real numbers. Unlike in the case of complex-analytic spaces, the structure sheaves of real-analytic spaces need not be coherent (cf. Coherent sheaf). Real-analytic spaces are said to be coherent if their structure sheaves are coherent. All real-analytic manifolds (i.e. smooth real-analytic spaces) are coherent real-analytic spaces.
Let be the germ at a point of a real-analytic subset of (cf. Analytic set). This defines the germ at of a complex-analytic subset of the space with the following equivalent properties: 1) is the intersection of all germs of complex-analytic sets containing ; 2) if is the analytic algebra of the germ , then is the analytic algebra of the germ . The germ is said to be the complexification of the germ , and is said to be the real part of the germ . Analogously, for any coherent real-analytic countably-infinite space it is possible to construct the complexification , which is a complex-analytic space. will then have a fundamental system of neighbourhoods in which are Stein spaces (cf. Stein space).
The theory of coherent real-analytic spaces is similar to the theory of complex Stein spaces. Global sections of any coherent analytic sheaf of modules on a coherent real-analytic countably-infinite space generate modules of germs of its sections at any point of , and all groups vanish if .
For any finite-dimensional coherent real-analytic countably-infinite space there exists a morphism
such that is a proper one-to-one mapping of into a coherent subspace in , while is an imbedding at the smooth points of . In particular, any (Hausdorff and countably-infinite) real-analytic manifold is isomorphic to a real-analytic submanifold in . For a reduced coherent real-analytic space the set of classes of isomorphic real-analytic principal fibre bundles with a real structure Lie group , admitting complexification, and base is in one-to-one correspondence with the set of classes of isomorphic topological principal fibre bundles with the same structure group .
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Real-analytic space. D.A. Ponomarev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Real-analytic_space&oldid=14527