mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the ''n''-dimensional complex coordinate space (or complex ''n''-space) is the set of all ordered ''n''-

tuple In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...

s of

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s, also known as ''

complex vector In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''. The operations of vector addition and ...

s''. The space is denoted

\Complex^n

, and is the ''n''-fold

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

of the

complex line In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line has complex dimension one over C (hence the term " line"), it has ordinary dime ...

\Complex

with itself. Symbolically,

\Complex^n = \left\

\Complex^n = \underbrace_.

The variables

z_i

are the (complex) coordinates on the complex ''n''-space. The special case

\Complex^2

, called the ''complex coordinate plane'', is not to be confused with the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

, a graphical representation of the complex line. Complex coordinate space is a

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

over the complex numbers, with componentwise addition and

scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...

. The real and imaginary parts of the coordinates set up a

bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

\Complex^n

with the 2''n''-dimensional

real coordinate space In mathematics, the real coordinate space or real coordinate ''n''-space, of dimension , denoted or , is the set of all ordered -tuples of real numbers, that is the set of all sequences of real numbers, also known as '' coordinate vectors''. ...

\mathbb R^

. With the standard

Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative function \, \cdot ...

\Complex^n

is a

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

over the complex numbers. A function on an open subset of complex ''n''-space is holomorphic if it is

holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...

in each complex coordinate separately.

Several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...

is the study of such holomorphic functions in ''n'' variables. More generally, the complex ''n''-space is the target space for holomorphic coordinate systems on

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

References

* {{citation, author1-link = Robert Gunning (mathematician), first = Robert , last = Gunning, author2=Hugo Rossi , title=Analytic functions of several complex variables Several complex variables Topological vector spaces

See also

References