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In the [[Mathematics|mathematical]] study of [[Several complex variables|several complex variables]], the '''Bergman kernel''', named after [[Biography:Stefan Bergman|Stefan Bergman]], is the reproducing kernel for the [[Hilbert space]] ([[Reproducing kernel Hilbert space|RKHS]]) of all square integrable [[Holomorphic function|holomorphic function]]s on a domain ''D'' in&nbsp;'''C'''<sup>''n''</sup>.
In detail, let [[Lp space|L<sup>2</sup>(''D'')]] be the Hilbert space of square integrable functions on ''D'', and let ''L''<sup>2,''h''</sup>(''D'') denote the subspace consisting of holomorphic functions in L<sup>2</sup>(''D''): that is,
:<math>L^{2,h}(D) = L^2(D)\cap H(D)</math>
where ''H''(''D'') is the space of holomorphic functions in ''D''.  Then ''L''<sup>2,''h''</sup>(''D'') is a Hilbert space: it is a [[Closed set|closed]] linear subspace of ''L''<sup>2</sup>(''D''), and therefore [[Complete metric space|complete]] in its own right.  This follows from the fundamental estimate, that for a holomorphic square-integrable function ''&fnof;'' in ''D''
{{NumBlk|:|<math>\sup_{z\in K} |f(z)| \le C_K\|f\|_{L^2(D)}</math>|{{EquationRef|1}}}}
for every compact subset ''K'' of ''D''.  Thus convergence of a sequence of holomorphic functions in ''L''<sup>2</sup>(''D'') implies also [[Compact convergence|compact convergence]], and so the limit function is also holomorphic.

Another consequence of ({{EquationRef|1}}) is that, for each ''z''&nbsp;&isin;&nbsp;''D'', the evaluation
:<math>\operatorname{ev}_z : f\mapsto f(z)</math>
is a continuous linear functional on ''L''<sup>2,''h''</sup>(''D'').  By the [[Riesz representation theorem]], this functional can be represented as the inner product with an element of  ''L''<sup>2,''h''</sup>(''D''), which is to say that
:<math>\operatorname{ev}_z f = \int_D f(\zeta)\overline{\eta_z(\zeta)}\,d\mu(\zeta).</math>
The Bergman kernel ''K'' is defined by
:<math>K(z,\zeta) = \overline{\eta_z(\zeta)}.</math>
The kernel ''K''(''z'',&zeta;) is holomorphic in ''z'' and antiholomorphic in &zeta;, and satisfies
:<math>f(z) = \int_D K(z,\zeta)f(\zeta)\,d\mu(\zeta).</math>

One key observation about this picture is that ''L''<sup>2,''h''</sup>(''D'') may be identified with the space of  <math>L^2</math> holomorphic (n,0)-forms on D,  via multiplication by  <math>dz^1\wedge \cdots \wedge dz^n</math>. Since the <math>L^2</math> inner product on this space is manifestly invariant under biholomorphisms of D, the Bergman kernel and the associated [[Bergman metric]] are therefore automatically invariant under the automorphism group of the domain.

==See also==
* [[Bergman metric]]
* [[Bergman space]]
* [[Szegő kernel]]

==References==
* {{Citation | last1=Krantz | first1=Steven G. | title=Function Theory of Several Complex Variables | publisher=American Mathematical Society | location=Providence, R.I. | isbn=978-0-8218-2724-6 | year=2002}}.
* {{springer|title=Bergman kernel function|id=B/b015560|first=E.M.|last=Chirka}}.

[[Category:Several complex variables]]

{{Sourceattribution|Bergman kernel}}