a locally bounded family

A locally bounded family is a collection of functions that, while it need not share a single ceiling across the entire domain, does share a ceiling on every compact (closed-and-bounded) inner piece. Picture a domain whose functions might shoot up toward the boundary — that is allowed — as long as, on any fixed inner patch, ALL the functions in the family stay under one common bar. It is the precise hypothesis that powers Montel's theorem.

Precisely, a family F of functions on an open region D is locally bounded if for every compact subset K of D there exists a single constant M (which may depend on K, but not on the function) such that |f(z)| <= M for all z in K and all f in F. The decisive feature is that the bound is uniform over the family on each compact set: it is one M for everybody, not a separate bound per function. Equivalently, every point of D has a neighborhood on which all functions of F are simultaneously bounded. Note the contrast with 'globally bounded' (one M for the whole of D): local boundedness is weaker and far more common, since it tolerates blow-up near the edge of D.

This is the input to Montel's theorem: a locally bounded family of HOLOMORPHIC functions is normal. The reason local boundedness is the right hypothesis is that, for holomorphic functions, the Cauchy estimates upgrade a bound on |f| over a disk into a bound on |f'| over a slightly smaller disk, and a uniform derivative bound is equicontinuity — feeding straight into Arzela-Ascoli. Honest caveat: for general continuous functions, local boundedness does NOT imply normality (you also need equicontinuity, which can fail). The implication 'locally bounded -> normal' is special to holomorphy, where the derivative is controlled by the function's size.