Oka–Weil Theorem
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In mathematics, especially the theory of
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 ...
, 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 as the function domain i ...
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 de ...
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 one of the most influential mathematicians of the twentieth century. His influence is du ...
.


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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
\mathcal(X)-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 \mathcal(X) (in particular, by polynomials).


Applications

Since Runge's theorem 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 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_ of holomorphic functions on \mathbb^n (and subsequently the sheaf \mathcal_ of holomorphic functions on a complex manifold X) is coherent.In paper it was call ...


References


Bibliography

* * * * * *


Further reading

* – An example where Runge's theorem does not hold. * Several complex variables Theorems in complex analysis {{mathanalysis-stub