Do complex polynomials form an orthonormal basis in the complex functional ($L^2$) space 
Let $D:=\{z\in C:|z|<1\}$ and consider polynomials $p_n:D\to C,p_n(z):=\sqrt{\frac{n+1}{\pi}}z^n,n\in N_0$.
Is $\{p_n\}$ an orthonormal basis for $L^2(D)?$

It seems that they are orthonormal, but do they form a complete system? For real functions it is true, if I remember correctly.

Let $c=\frac I {m(D)}$ where $I$ is the integral of $|z|^{2}$ over $D$. Then $|z|-c$ is orthogonal to each $p_n$ and hence the sequence is not a basis. 
