This is a follow-up to my question here. Let $D$ be the unit disk, and for each $n$ let $f_n\in L^2(D)$ be a polynomial in $z=x+iy$ with complex coefficients. And suppose that $f_n\rightarrow f$ with respect to the $L^2(D)$ norm for some $f\in L^2(D)$. My question is, is it necessarily true that $f$ is holomorphic?
If not, does anyone know of a counterexample? I ask because this is true for uniform convergence.
If $c \in D$ and $B$ is a disk around $c$ whose closure is contained in $D$ then $f_n(c)-f_m(c)$ is the average of $f_n-f_m$ over $B$. Together with Cauchy - Schwarz inequality this tells you that $\{f_n\}$ actually converges uniformly on $B$. If a sequence of holomorphic functions converges locally uniformly the limit is holomorphic. Note that $f$ is only an equivalence class of functions, so the correct statement is $f=g$ almost everywhere with $g$ holomorphic.
