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Normal family

In mathematics, with special application to complex analysis, a normal family is a pre-compact family of continuous functions. Informally, this means that the functions in the family are not exceedingly numerous or widely spread out, rather, they stick together in a relatively "compact" manner. It is of general interest to understand compact sets in function spaces, since these are usually truly infinite-dimensional in nature.

More formally, a set (sometimes called a family) F of continuous functions f defined on some complete metric space X with values in another complete metric space Y is called normal if every sequence of functions in F contains a subsequence which converges uniformly on compact subsets of X to a continuous function from X to Y.

This definition is often used in complex analysis for spaces of holomorphic functions. It turns out that a sequence of holomorphic functions, that converges uniformly on compact sets, converges to a holomorphic function. So you can replace X with a region in the complex plane, Y with the complex plane itself and every instance of continuous with holomorphic and you get a version of the definition most used in complex analysis.

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

$f_n(z) \to f(z)$

uniformly on compact subsets to mean that

$d\left(f_n(z),f(z)\right)\,$

goes to 0 uniformly on compact subsets.

Note that this is a classical definition that, while very often used, is not really consistent with modern naming. In more modern language, one would give a metric on the space of continuous (holomorphic) functions that corresponds to convergence on compact subsets and then you would say "precompact set of functions" in such a metric space instead of saying "normal family of continuous (holomorphic) functions". This added generality however makes it more cumbersome to use since one would need to define the metric mentioned above.

References

John B. Conway. (1978) Functions of One Complex Variable I. Springer-Verlag, New York, New York.

Last updated: 10-10-2005 09:31:47
03-10-2013 05:06:04
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