= Bergman space
{c}
{title2=$A^2(D)$}
For a planar domain $D$, the Bergman space $A^2(D)$ consists of <holomorphic functions> with finite area <L2 norm>, using the inherited <inner product>. It is a closed subspace of $L^2(D)$. Indeed the disk mean-value estimate $|f(z)|\leq(\pi r^2)^{-1/2}\|f\|_{L^2(B(z,r))}$ bounds values locally by the <L2 norm>. An $L^2$-Cauchy sequence of holomorphic functions consequently converges uniformly on compact subsets to a holomorphic function, agreeing almost everywhere with its $L^2$ limit.
On the unit disk, the normalized monomials $\sqrt{(n+1)/\pi}\,z^n$, $n\geq0$, form an <orthonormal basis>. Angular integration makes them orthogonal and gives $\|z^n\|_2^2=\pi/(n+1)$. Applying the same integration to a Taylor expansion gives $\|\sum_na_nz^n\|_2^2=\pi\sum_n|a_n|^2/(n+1)$, proving completeness of this basis.
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