# Are polynomials dense is L2 of the unit disk?

### Solution:

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.