Are polynomials dense is L2 of the unit disk?

Question

Let $D$ be the unit disk in the complex plane, and let $X$ be the subset of $L^2(D)$ consisting of polynomials in the complex variable $z=x+iy$ with complex coefficients. My question is, is $X$ dense in $L^2(D)$?

If not, does anyone know of a function in $L^2(D)$ which cannot be written as an $L^2$ limit of polynomials?

Answer 1

No. For instance, $g(z)=\overline{z}$ is orthogonal to every polynomial (proof sketch: if $f(z)=z^n$, then $\langle f,g\rangle=\int_D f\overline{g}=\int_D z^{n+1}=0$ by either direct computation in polar coordinates or using the symmetries of $z^{n+1}$). It follows that $g$ is not in the closure of the polynomials.

Answer 2

Yes. $X$ is dense in $L^2$.

Certainly $X$ is dense in the set of all the continuous functions, $C(D)$. $C(D)$ is also dense in $L^2$, so $X$ is dense in $L^2$.