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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 A is the space of all holomorphic functions f {\displaystyle f} in D for which the p-norm is finite:
The quantity ‖ f ‖ A p {\displaystyle \|f\|_{A^{p}}} is called the norm of the function f; it is a true norm if p ≥ 1 {\displaystyle p\geq 1}. Thus A is the subspace of holomorphic functions that are in the space L. 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 L implies also compact convergence, and so the limit function is also holomorphic.
If p = 2, then A is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.