a normal family

A normal family is a collection of holomorphic functions that is 'compact' in a useful sense: no matter which infinite sequence you pull out of it, you can always extract a subsequence that converges nicely. It is the function-space analogue of the Bolzano-Weierstrass property for numbers, where every bounded sequence has a convergent subsequence. Compactness is the great problem-solving lever in analysis — it lets you take a limit and know the limit is still in the game — and a normal family is precisely the setting where that lever exists for holomorphic functions.

Precisely, a family F of holomorphic functions on an open region D is called normal if every sequence f_n drawn from F has a subsequence that converges locally uniformly on D — where the limit is allowed either to be a holomorphic function, or (in the broader convention used in dynamics) to be the constant infinity, meaning the subsequence escapes uniformly to infinity on compacta. The crucial point is that you do not need the original sequence to converge; you only need to be able to FIND a convergent subsequence inside it. By the Weierstrass convergence theorem any such limit is again holomorphic, so a normal family is closed under taking these subsequential limits — it is a self-contained, compact world of functions.

Why care? Normality turns existence questions into the line 'extract a convergent subsequence and pass to the limit'. The Riemann mapping theorem is proved exactly this way: you set up a normal family of candidate maps, take a sequence whose derivatives at the basepoint approach the largest possible value, extract a locally uniform limit, and show that limit is the desired conformal map. Montel's theorem is the workhorse that recognizes normal families in practice (local boundedness is enough). Honest caveat: normality is a property of the FAMILY, not of any one function, and the convergent subsequence's limit can differ from sequence to sequence — there is no claim that the whole family converges to anything.