complex lineComplexLine2004-07-23 16:05:462005-03-05 12:25:57Definitioncomplex affine space% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{defn}{Definition}\begin{defn}
Let $a, b \in {\mathbb{C}}^n$. The set
$\ell := \{a + b z \mid z \in {\mathbb{C}} \}$ is called the {\em complex line}.
\end{defn}
A complex line is a holomorphic complex affine imbedding of ${\mathbb{C}}$
into ${\mathbb{C}}^n$ so that if $f$ is holomorphic, then
$z \mapsto f(a + b z)$ is also holomorphic. That is the complex structures of $\ell$ and ${\mathbb{C}}^n$ are compatible. Hence not every two dimensional real affine space is a complex line.
\begin{defn}
Let $a, b_1, \ldots, b_k \in {\mathbb{C}}^n$ such that
$b_1, \ldots, b_k$ are linearly independent
over ${\mathbb{C}}$,
then. The set
\begin{equation*}
\{a + \sum_{j=1}^k b_k z_k \mid z_1,\ldots,z_k \in {\mathbb{C}} \}
\end{equation*}
is called the {\em complex affine space}.
\end{defn}
\begin{thebibliography}{9}
\bibitem{Krantz:several}
Steven~G.\@ Krantz.
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}},
AMS Chelsea Publishing, Providence, Rhode Island, 1992.
\end{thebibliography}