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| In the [[mathematics|mathematical]] study of [[several complex variables]], the '''Bergman kernel''', named after [[Stefan Bergman]], is a [[reproducing kernel]] for the [[Hilbert space]] of all [[square integrable]] [[holomorphic function]]s on a domain ''D'' in '''C'''<sup>''n''</sup>.
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| 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 ''D'': that is,
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| :<math>L^{2,h}(D) = L^2(D)\cap H(D)</math>
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| 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 ''ƒ'' in ''D''
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| {{NumBlk|:|<math>\sup_{z\in K} |f(z)| \le C_K\|f\|_{L^2(D)}</math>|{{EquationRef|1}}}}
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| for every [[compact set|compact]] subset ''K'' of ''D''. Thus convergence of a sequence of holomorphic functions in ''L''<sup>2</sup>(''D'') implies also [[compact convergence]], and so the limit function is also holomorphic.
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| Another consequence of ({{EquationRef|1}}) is that, for each ''z'' ∈ ''D'', the evaluation
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| :<math>\operatorname{ev}_z : f\mapsto f(z)</math>
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| 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
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| :<math>\operatorname{ev}_z f = \int_D f(\zeta)\overline{\eta_z(\zeta)}\,d\mu(\zeta).</math>
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| The Bergman kernel ''K'' is defined by
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| :<math>K(z,\zeta) = \overline{\eta_z(\zeta)}.</math>
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| The kernel ''K''(''z'',ζ) holomorphic in ''z'' and antiholomorphic in ζ, and satisfies
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| :<math>f(z) = \int_D K(z,\zeta)f(\zeta)\,d\mu(\zeta).</math>
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| ==See also==
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| * [[Bergman metric]]
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| * [[Bergman space]]
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| * [[Szegő kernel]]
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| ==References==
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| * {{Citation | last1=Krantz | first1=Steven G. | authorlink=Steven Krantz|title=Function Theory of Several Complex Variables | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-2724-6 | year=2002}}.
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| * {{springer|title=Bergman kernel function|id=B/b015560|first=E.M.|last=Chirka}}.
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| [[Category:Several complex variables]]
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