# Bergman space

In complex analysis, a branch of mathematics, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, ${\displaystyle A^{p}(D)}$ is the space of holomorphic functions in D such that the p-norm

${\displaystyle \|f\|_{p}=\left(\int _{D}|f(x+iy)|^{p}\,dx\,dy\right)^{1/p}<\infty .}$

Thus ${\displaystyle A^{p}(D)}$ is the subspace of holomorphic functions that are in the space Lp(D). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D: Template:NumBlk Thus convergence of a sequence of holomorphic functions in Lp(D) implies also compact convergence, and so the limit function is also holomorphic.

If p = 2, then ${\displaystyle A^{p}(D)}$ is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.

## References

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