Bergman space

Let $G\subset {ℂ}^n$ be a domain and let$dV$ denote the Euclidean volume measure on $G$.

Definition.

Let

$A^2(G)$ is called the Bergman space on $G$. The norm on this spaceis defined as

$$\parallel f\parallel :=\sqrt{{\int }_G{|f(z)|}^2𝑑V(z)}.$$

Further we define an inner product on $A^2(G)$ as

$$⟨f,g⟩:={\int }_{G}f(z)\overline{g(z)}𝑑V(z).$$

The inner product as defined above really is an inner product and furtherit can be shown that $A^2(G)$ is complete since convergence in the above norm implies normal convergence (uniform convergence oncompact subsets). The space $A^2(G)$ is therefore a Hilbert space.Sometimes this space is also denoted by $L_a^2(G)$.