# Bergman space

Let $G\subset {\u2102}^{n}$ 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 space is defined as

$$\parallel f\parallel :=\sqrt{{\int}_{G}{|f(z)|}^{2}\mathit{d}V(z)}.$$ |

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

$$\u27e8f,g\u27e9:={\int}_{G}f(z)\overline{g(z)}\mathit{d}V(z).$$ |

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 |
---|---|

Canonical name | BergmanSpace |

Date of creation | 2013-03-22 15:04:43 |

Last modified on | 2013-03-22 15:04:43 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 10 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32A36 |

Related topic | BergmanKernel |