Bergman space ($A^2(D)$)

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 $|f(z)|\le (\pi r^2{)}^{-1/2}\Vert f{\Vert }_{L^2(B(z,r))}$ 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 $\sqrt{(n+1)/\pi }{z}^n$, $n\ge 0$, form an orthonormal basis. Angular integration makes them orthogonal and gives $\Vert z^n{\Vert }_2^2=\pi /(n+1)$. Applying the same integration to a Taylor expansion gives $\Vert {\sum }_n{a}_n{z}^n{\Vert }_2^2=\pi {\sum }_n|a_n{|}^2/(n+1)$, proving completeness of this basis.

In the area Hilbert space of the unit disk, conjugates of Bergman space functions vanishing at zero are orthogonal to all holomorphic functions, since ${\int }_{D}hgdA=\pi h(0)g(0)$. This follows from the angular mean-value property and the Cauchy-Schwarz inequality, which makes the product integrable. But the radial function $|z{|}^2-1/2$ is also orthogonal to every holomorphic function: its radial integral against the constant angular mean is zero. It is not conjugate holomorphic, since it is real-valued and nonconstant. Thus the conjugate-holomorphic family is a proper subset of the orthogonal complement.