# Complex space

complex-analytic space

An analytic space over the field of complex numbers $\mathbf C$. The simplest and most widely used complex space is the complex number space $\mathbf C ^ {n}$. Its points, or elements, are all possible $n$- tuples $( z _ {1} \dots z _ {n} )$ of complex numbers $z _ \nu = x _ \nu + iy _ \nu$, $\nu = 1 \dots n$. It is a vector space over $\mathbf C$ with the operations of addition

$$z + z ^ \prime = \ ( z _ {1} + z _ {1} ^ \prime \dots z _ {n} + z _ {n} ^ \prime )$$

and multiplication by a scalar $\lambda \in \mathbf C$,

$$\lambda z = \ ( \lambda z _ {1} \dots \lambda z _ {n} ),$$

as well as a metric space with the Euclidean metric

$$\rho ( z, z ^ \prime ) = \ | z - z ^ \prime | = \ \sqrt {\sum _ {\nu = 1 } ^ { n } | z _ \nu - z _ \nu ^ \prime | ^ {2} } =$$

$$= \ \sqrt {\sum _ {\nu = 1 } ^ { n } ( x _ \nu - x _ \nu ^ \prime ) ^ {2} + ( y _ \nu - y _ \nu ^ \prime ) ^ {2} } .$$

In other words, the complex number space $\mathbf C ^ {n}$ is obtained as the result of complexifying the real number space $\mathbf R ^ {2n}$. The complex number space $\mathbf C ^ {n}$ is also the topological product of $n$ complex planes $\mathbf C ^ {1} = \mathbf C$, $\mathbf C ^ {n} = \mathbf C \times \dots \times \mathbf C$.

How to Cite This Entry:
Complex space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complex_space&oldid=46431
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article