In mathematics, especially the theory of
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
, the Oka–Weil theorem is a result about the
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
of
holomorphic function
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 derivativ ...
s on
Stein spaces due to
Kiyoshi Oka
was a Japanese mathematician who did fundamental work in the theory of several complex variables.
Biography
Oka was born in Osaka. He went to Kyoto Imperial University in 1919, turning to mathematics in 1923 and graduating in 1924.
He was in ...
and
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
.
Statement
The Oka–Weil theorem states that if ''X'' is a Stein space and ''K'' is a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
-convex subset of ''X'', then every holomorphic function in an
open neighborhood
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a po ...
of ''K'' can be approximated uniformly on ''K'' by holomorphic functions on
(i.e. by polynomials).
Applications
Since
Runge's theorem
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following:
Denoting by C the set of complex numbers, let ''K ...
may not hold for several complex variables, the Oka–Weil theorem is often used as an approximation theorem for several complex variables. The
Behnke–Stein theorem
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence G_k \subset \mathbb^n (i.e., G_k \subset G_) of domains of holomorphy is again a domain of holomorphy. It was proved by ...
was originally proved using the Oka–Weil theorem.
See also
*
Oka coherence theorem
In mathematics, the Oka coherence theorem, proved by , states that the sheaf \mathcal := \mathcal_ of germs of holomorphic functions on \mathbb^n over a complex manifold is coherent.In paper it was called the idéal de domaines indéterminés.
...
References
Bibliography
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Further reading
* – An example where
Runge's theorem
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following:
Denoting by C the set of complex numbers, let ''K ...
does not hold.
*
Several complex variables
Theorems in complex analysis
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