# Bergman space

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

###### Definition.

Let

 $A^{2}(G):=\Big{\{}f\text{ holomorpic in }G~{}\Big{|}~{}\sqrt{\int_{G}\lvert f(% z)\rvert^{2}dV(z)}<\infty\Big{\}}.$

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

 $\lVert f\rVert:=\sqrt{\int_{G}\lvert f(z)\rvert^{2}dV(z)}.$

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

 $\langle f,g\rangle:=\int_{G}f(z)\overline{g(z)}dV(z).$

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

## References

• 1 D’Angelo, John P. , CRC Press, 1993.
• 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Bergman space BergmanSpace 2013-03-22 15:04:43 2013-03-22 15:04:43 jirka (4157) jirka (4157) 10 jirka (4157) Definition msc 32A36 BergmanKernel