Difference between revisions of "Complex space"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
||
Line 18: | Line 18: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) {{MR|0205211}} {{MR|0205210}} {{ZBL|0301.54002}} {{ZBL|0301.54001}} {{ZBL|0145.19302}} </TD></TR></table> |
Line 26: | Line 26: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German) {{MR|0580152}} {{ZBL|0433.32007}} </TD></TR></table> |
Revision as of 21:51, 30 March 2012
complex-analytic space
An analytic space over the field of complex numbers . The simplest and most widely used complex space is the complex number space
. Its points, or elements, are all possible
-tuples
of complex numbers
,
. It is a vector space over
with the operations of addition
![]() |
and multiplication by a scalar ,
![]() |
as well as a metric space with the Euclidean metric
![]() |
![]() |
In other words, the complex number space is obtained as the result of complexifying the real number space
. The complex number space
is also the topological product of
complex planes
,
.
References
[1] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001 |
[2] | N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) MR0205211 MR0205210 Zbl 0301.54002 Zbl 0301.54001 Zbl 0145.19302 |
Comments
A more general notion of complex space is contained in [a1]. Roughly it is as follows. Let be a Hausdorff space equipped with asheaf
of local
-algebras (a so-called
-algebraized space). Two such spaces
and
are called isomorphic if there is a homeomorphism
and a sheaf isomorphism
(cf. [a1]). Now, a
-algebraized space
is called a complex manifold if it is locally isomorphic to a standard space
,
a domain,
its sheaf of germs of holomorphic functions, i.e. if for every
there is a neighbourhood
of
in
and a domain
, for some
, so that the
-algebraized spaces
and
are isomorphic. Let
be a domain and
a coherent ideal. The support
of the (coherent) quotient sheaf
is a closed set in
, and the sheaf
is a (coherent) sheaf of local
-algebras. The
-algebraized space
is called a (closed) complex subspace of
(it is naturally imbedded in
via the quotient sheaf mapping). A complex space
is a
-algebraized space that is locally isomorphic to a complex subspace, i.e. every point
has a neighbourhood
so that
is isomorphic to a complex subspace of a domain in some
. (See also Sheaf theory; Coherent sheaf.) More on complex spaces, in particular their use in function theory of several variables and algebraic geometry, can be found in [a1]. See also Stein space; Analytic space.
References
[a1] | H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German) MR0580152 Zbl 0433.32007 |
Complex space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complex_space&oldid=11574