Properties of C(G) and H(G)

Abstract

With little change we can study functions of several variables. To keep the notation simple, we will restrict ourselves to two variables. In that case, G is an open set in ℂ × ℂ = ℂ². Then C(G) is defined just as in the one-variable case. (The distance in ℂ² between two points (z,w) and (z’,w’) will be denoted d((z,w), (z’,w’)) = (|z - z’|² + |w - w’|²)^½. We also define H(G) as before, but must first define holomorphic.