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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, $$A^p(D)$$ is the space of holomorphic functions in D such that the p-norm

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

Thus $$A^p(D)$$ is the subspace of homolorphic 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:

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

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 $$A^p(D)$$ is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.
References

Richter, Stefan (2001), "Bergman spaces", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104.

Mathematics Encyclopedia