Let $F$ be the set analytic functions in $B(0,1)$ satisfying $f(0)=0$, $f'(0)=1$. Prove $F$ is not normal.
I've been trying to come up with a counter example, but I am failing. Does anyone have a suggestion on how I can show this?
The family of functions F is said to be normal in a subset of $S\subset \mathbb{C}$ if every subsequence $\{f_n\}$ of functions $f_n\in F$ contains a subsequence which converges uniformly on every compact subset of S.
Let $f_n (z)=z+nz^{2}$. Then $f_n (0)=0$, $f_n'(0)=1$ but no subsequence converges uniformly on compact subsets. For example, $[0,1/2]$ is a compact subset on which the sequence is unbounded.