|
|
|
Viewing Version
1
of
'holomorphically convex'
|
[ view 'holomorphically convex'
|
back to history
]
| Title of object: |
holomorphically convex |
| Canonical Name: |
HolomorphicallyConvex |
| Type: |
Definition |
| Created on: |
2005-02-22 12:54:35 |
| Modified on: |
2005-02-22 12:54:35 |
| Classification: |
msc:32E05 |
| Defines: |
holomorphically convex hull |
| Synonyms: |
holomorphically convex=holomorph-convex |
Revision comment (for changes between this and next version):
| Changes for correction #5970 ('use of domain'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
\theoremstyle{definition}
\newtheorem*{defn}{Definition}
Let ${\mathcal{O}}(G)$ stand for the set of functions holomorphic on a domain $G$.
\begin{defn}
Let $K \subset G \subset {\mathbb{C}}^n$ be a compact set, where $G$ is an open domain.
We define the {\em holomoprhically \PMlinkescapetext{convex hull}} of $K$ as
\begin{equation*}
\hat{K}_G := \{ z \in G \mid \lvert f(z) \rvert \leq \sup_{w \in K} \lvert f(w) \rvert \text{ for all } f \in {\mathcal{O}}(G) \} .
\end{equation*}
The domain $G$ is called {\em holomorhpically convex} if for every $K \subset G$ compact in $G$, $\hat{K}_G$ is also compact in $G$.
\end{defn}
Note that when $n=1$, any domain $G$ is holomorphically convex. Also note that this is the same as being a domain of holomorphy.
The above definition is also used when $G$ is not a domain of ${\mathbb{C}}^n$
but is in fact a complex analytic manifold. All that is required is then to remove the ${\mathbb{C}}^n$ from the definition.
\begin{thebibliography}{9}
\bibitem{Hormander:several}
Lars H\"ormander.
{\em \PMlinkescapetext{An Introduction to Complex Analysis in Several
Variables}},
North-Holland Publishing Company, New York, New York, 1973.
\bibitem{Krantz:several}
Steven~G.\@ Krantz.
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}},
AMS Chelsea Publishing, Providence, Rhode Island, 1992.
\end{thebibliography} |
|
|
|
|
|
