Oka–Weil theorem

In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil.

Statement

The Oka–Weil theorem states that if X is a Stein space and K is a compact ${\displaystyle {\mathcal {O}}(X)}$-convex subset of X, then every holomorphic function in an open neighborhood of K can be approximated uniformly on K by holomorphic functions on ${\displaystyle {\mathcal {O}}(X)}$ (in particular, by polynomials).^[1]

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