Bergman space

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^p(D) is the space of all holomorphic functions ${\displaystyle f}$ in D for which the p-norm is finite:

The quantity ${\displaystyle \|f\|_{A^{p}(D)}}$ is called the norm of the function f; it is a true norm if ${\displaystyle p\geq 1}$. Thus A^p(D) is the subspace of holomorphic functions that are in the space L^p(D). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D:

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