In complex analysis, functional analysis and operator theory, a Bergman space 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, for 0 < p < ∞, the Bergman space Ap(D) is the space of all holomorphic functions in D for which the p-norm is finite:
The quantity is called the norm of the function f; it is a true norm if . Thus Ap(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:
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 Ap(D) is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.
Special cases and generalisations
If the domain D is bounded, then the norm is often given by
where is a normalised Lebesgue measure of the complex plane, i.e. dA = dz/Area(D). Alternatively dA = dz/π is used, regardless of the area of D.
The Bergman space is usually defined on the open unit disk of the complex plane, in which case . In the Hilbert space case, given , we have
that is, A2 is isometrically isomorphic to the weighted ℓp(1/(n+1)) space. In particular the polynomials are dense in A2. Similarly, if D = ℂ+, the right (or the upper) complex half-plane, then
where , that is, A2(ℂ+) is isometrically isomorphic to the weighted Lp1/t (0,∞) space (via the Laplace transform).
The weighted Bergman space Ap(D) is defined in an analogous way, i.e.
provided that w : D → [0, ∞) is chosen in such way, that is a Banach space (or a Hilbert space, if p = 2). In case where , by a weighted Bergman space  we mean the space of all analytic functions f such that
and similarly on the right half-plane (i.e. ) we have
and this space is isometrically isomorphic, via the Laplace transform, to the space , where
(here Γ denotes the Gamma function).
Further generalisations are sometimes considered, for example denotes a weighted Bergman space (often called a Zen space) with respect to a translation-invariant positive regular Borel measure on the closed right complex half-plane , that is
The reproducing kernel of A2 at point is given by
and similarly for we have
In general, if maps a domain conformally onto a domain , then
In weighted case we have
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