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Bergman space
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(Definition)
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Let
 be a domain and let  denote the Euclidean volume measure on  .
Definition 1 Let
 holomorpic in  |
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 is called the Bergman space on  . The norm on this space is defined as
Further we define an inner product on  as
The inner product as defined above really is an inner product and further it can be shown that  is complete since convergence in the above norm can be shown to be the same as normal convergence (uniform convergence on compact subsets). The space  is therefore a Hilbert space. Sometimes
this space is also denoted by  .
- 1
- D'Angelo, John P. Several complex variables and the geometry of real hypersurfaces, CRC Press, 1993.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"Bergman space" is owned by jirka.
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(view preamble)
Cross-references: Hilbert space, compact subsets, uniform convergence, normal convergence, complete, inner product, norm, euclidean volume measure, domain
There are 2 references to this entry.
This is version 6 of Bergman space, born on 2005-02-22, modified 2007-12-14.
Object id is 6801, canonical name is BergmanSpace.
Accessed 2533 times total.
Classification:
| AMS MSC: | 32A36 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Bergman spaces) |
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Pending Errata and Addenda
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