 Normal family

A family $\mathcal{F}$ of meromorphic functions on a domain is a normal family if every sequence from $\mathcal{F}$ has a subsequence converging uniformly on every compact subset, measured in the chordal metric on the Riemann sphere. The limit is a meromorphic function or the constant infinity. For holomorphic functions, a finite limit is holomorphic; a sequence may also tend locally uniformly to infinity.

The constant functions $f_n(z)=n$ form a normal family, because any sequence has either a constant subsequence or a subsequence tending uniformly to infinity in the chordal metric. In contrast $g_n(z)=nz$ is not a normal family on any neighbourhood of zero. Every subsequence tends to infinity at every nonzero point but remains zero at zero; this pointwise pattern cannot be a continuous Riemann sphere valued limit.

For a rational map $f$, the Fatou set is precisely where its iterations of a map form a normal family. Normality asks for convergent subsequences, so it allows stable rotations as well as convergence to an attracting periodic point.^[ref]

The term "normal family" typically refers to a family structure that aligns with widely accepted societal norms and expectations regarding family dynamics, roles, and relationships. However, the definition of what constitutes a "normal" family can vary greatly depending on cultural, social, and individual perspectives. In many Western cultures, a "normal family" often implies a nuclear family consisting of two parents (a mother and a father) and their biological children.