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In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is the reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain D in C.
In detail, let L be the Hilbert space of square integrable functions on D, and let L denote the subspace consisting of holomorphic functions in D: that is,
where H is the space of holomorphic functions in D. Then L is a Hilbert space: it is a closed linear subspace of L, and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in D
for every compact subset K of D. Thus convergence of a sequence of holomorphic functions in L implies also compact convergence, and so the limit function is also holomorphic.
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