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In the 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 functions on a domain D in Cn.

In detail, let L²(D) be the Hilbert space of square integrable functions on D, and let L^2,h(D) denote the subspace consisting of holomorphic functions in D: that is,

\( L^{2,h}(D) = L^2(D)\cap H(D) \)

where H(D) is the space of holomorphic functions in D. Then L^2,h(D) is a Hilbert space: it is a closed linear subspace of L²(D), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in D

\( \sup_{z\in K} |f(z)| \le C_K\|f\|_{L^2(D)}\) (1)

for every compact subset K of D. Thus convergence of a sequence of holomorphic functions in L2(D) implies also compact convergence, and so the limit function is also holomorphic.

Another consequence of (1) is that, for each z ∈ D, the evaluation

\( \operatorname{ev}_z : f\mapsto f(z)\)

is a continuous linear functional on L^2,h(D). By the Riesz representation theorem, this functional can be represented as the inner product with an element of L^2,h(D), which is to say that

\( \operatorname{ev}_z f = \int_D f(\zeta)\overline{\eta_z(\zeta)}\,d\mu(\zeta).\)

The Bergman kernel K is defined by

\( K(z,\zeta) = \overline{\eta_z(\zeta)}.\)

The kernel K(z,ζ) holomorphic in z and antiholomorphic in ζ, and satisfies

\( f(z) = \int_D K(z,\zeta)f(\zeta)\,d\mu(\zeta).\)