I am wondering about how to approach to this problem.
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ and $f_n:\mathbb{D}\to \mathbb{C}$ be a sequence of analytic functions with $||f_n||_2\leq C<\infty$ where $||.||_2$ is the usual $L^2$ norm on $\mathbb{D}$. Then prove that there is a subsequence of $f_n$ which converges uniformly compact subsets of $\mathbb{D}$ to an analytic function on $\mathbb{D}$.
I was wondering about the following :
How are the Lebesgue integral and usual complex integrals related? For real case (Riemann integration), we approximate our function $f$ by step functions, since they are simple functions hence we can relate between two integrals ( I know as far as the proper integrals are concerned, they are same). For complex case, for a curve $\gamma$, we take a partition of $\gamma$ and proceed in the same way. Will it be helpful to show that both integrals are identical?
Thank you for help in advance.
about the problem: one way is to find, for every $z\in\mathbb D$, a function $\phi_z$ on $\mathbb D$ such that for every holomorphic $f$ on $\mathbb D$ we have $$f(z)=\langle \phi_z,f\rangle.$$ For example, use the mean value property: choose a small closed disk $D_z\subset\mathbb D$ centered at $z$, say with radius $(1-|z|)/2$, and set $\phi_z=c_z^{-1}\chi_{D_z}$, where $c_z$ is area of $D_z$ ($\chi$ stands for characteristic function). We then have $\Vert\phi_z\Vert_2^2=c_z^{-1}$, hence $$|f_n(z)|\leq C\,{c_z}^{-1/2} = c\ (1-|z|)^{-1} $$ for every $n$ (for a suitable $c$).
And here we are - the functions are locally uniformly bounded, so they form a normal family (Montel's theorem).