Bergman space is a concept from functional analysis and complex analysis. It is named after the mathematician Stefan Bergman. Specifically, the Bergman space is a type of Hilbert space that consists of analytic functions defined on a domain in the complex plane, typically the unit disk or other bounded domains.
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For a planar domain , the Bergman space consists of holomorphic functions with finite area L2 norm, using the inherited inner product. It is a closed subspace of . Indeed the disk mean-value estimate bounds values locally by the L2 norm. An -Cauchy sequence of holomorphic functions consequently converges uniformly on compact subsets to a holomorphic function, agreeing almost everywhere with its limit.
On the unit disk, the normalized monomials , , form an orthonormal basis. Angular integration makes them orthogonal and gives . Applying the same integration to a Taylor expansion gives , proving completeness of this basis.