Complex Analysis
This definition is often used in complex analysis for spaces of holomorphic functions. As a consequence of Cauchy's integral theorem, a sequence of holomorphic functions that converges uniformly on compact sets must converge to a holomorphic function. Thus in complex analysis a normal family F of holomorphic functions in a region X of the complex plane with values in Y = C is such that every sequence in 'F' contains a subsequence which converges uniformly on compact subsets of X to a holomorphic function. Montel's theorem asserts that every locally bounded family of holomorphic functions is normal.
Another space where this is often used is the space of meromorphic functions. This is similar to the holomorphic case, but instead of using the standard metric (distance) for convergence we must use the spherical metric. That is if d is the spherical metric, then want
compactly to mean that
goes to 0 uniformly on compact subsets.
Read more about this topic: Normal Family
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