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2014-01-26T10:22:14Z

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In [[complex analysis]], a branch of [[mathematics]], a '''Bergman space''', named after [[Stefan Bergman]], is a [[function space]] of [[holomorphic function]]s in a [[Domain (mathematical analysis)|domain]] ''D'' of the [[complex plane]] that are sufficiently well-behaved at the boundary that they are absolutely [[integrable]]. Specifically, <math>A^p(D)</math> is the space of holomorphic functions in ''D'' such that the [[norm (mathematics)|p-norm]]

:<math>\|f\|_p = \left(\int_D |f(x+iy)|^p\,dx\,dy\right)^{1/p} < \infty.</math>

Thus <math>A^p(D)</math> is the subspace of holomorphic functions that are in the space [[Lp space|L<sup>''p''</sup>(''D'')]]. The Bergman spaces are [[Banach space]]s, which is a consequence of the estimate, valid on [[compact space|compact]] subsets ''K'' of ''D'':
{{NumBlk|:|<math>\sup_{z\in K} |f(z)| \le C_K\|f\|_{L^p(D)}.</math>|{{EquationRef|1}}}}
Thus convergence of a sequence of holomorphic functions in ''L''<sup>''p''</sup>(''D'') implies also [[compact convergence]], and so the limit function is also holomorphic.

If ''p'' = 2, then <math>A^p(D)</math> is a [[reproducing kernel Hilbert space]], whose kernel is given by the [[Bergman kernel]].

==References==

{{reflist}}
*{{citation|last=Bergman|first= Stefan|title=The kernel function and conformal mapping|edition=2nd|series=Mathematical Surveys|volume=5| publisher=American Mathematical Society|year= 1970}}
*{{springer|title=Bergman spaces|id=B/b120130|first=Stefan|last=Richter}}.
*{{citation|last=Hedenmalm|first= H.|last2= Korenblum|first2= B.|last3= Zhu |first3=K.|title=Theory of Bergman Spaces|year=2000|publisher=Springer|isbn=978-0-387-98791-0|url=http://www.springer.com/mathematics/analysis/book/978-0-387-98791-0}}

[[Category:Complex analysis]]

{{mathanalysis-stub}}

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