Normal family

Remarks

Example: $\mathcal{F} = \set{z^n : n \ge 1}$ on the unit disc $\mathbb{D}$ is normal. On a compact set $K \subset \mathbb{D}$ there is $r < 1$ with $\abs{z} \le r$ on $K$, so $\abs{z^n} \le r^n \to 0$ uniformly on $K$. Non-example: $\set{n z : n \ge 1}$ on $\mathbb{D}$ is not normal. Each subsequence tends to $\infty$ at each $z \neq 0$ but stays $0$ at $z = 0$, so it does not converge on a disc around $0$. The limit of a convergent subsequence is holomorphic, by Weierstrass's theorem that uniform limits of holomorphic functions are holomorphic. Many authors also accept a subsequence that tends to $\infty$ uniformly on compact subsets. With this convention, and with the chordal metric of the Riemann sphere, the definition extends to families of meromorphic functions.