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### Problem 1

Let $V\subset \mathbb C$ be open and connected and $\mathcal F\subset H(V)$. If $\mathcal F$ is a normal family, then $\mathcal F' = \{f': f\in\mathcal F\}$ is also a normal family.

### Problem 2

1. The converse to the result in Problem 1 is false.
2. Find a "small" hypothesis to make the converse true.

### Problem 3

Let $V\subset \mathbb C$ be open and connected and $\mathcal F\subset H(V)$ a normal family. Let $f_n\in \mathcal F$ a sequence such that $f_n(z)\to f(z)$ for each $z\in S\subset V$, where $S$ has a limit point in $V$ and $f\in H(V)$. Then $f_n\to f$ in $H(V)$.

### Problem 4

Let $V\subset \mathbb C$ be open, $M>0$, and
$\displaystyle\mathcal F = \Big\{ f\in H(V): \iint_V |f(z)|^2 dxdy \le M \Big\}$.
Then $\mathcal F$ is a normal family.

### Problem 5

Let $V\subset \mathbb C$ be open and connected and $\{f_n\}\subset H(V)$ a normal family such that each $f_n$ is never zero in $V$. If there exists $z\in V$ such that $f_n(z)\to 0$, then $f_n\to 0$ in $H(V)$.