Continuous mappings | Complex analysis | Topology of function spaces

Normal family

In mathematics, with special application to complex analysis, a normal family is a pre-compact subset of the space of continuous functions. Informally, this means that the functions in the family are not widely spread out, but rather stick together in a somewhat "clustered" manner. Note that a compact family of continuous functions is automatically a normal family.Sometimes, if each function in a normal family F satisfies a particular property (e.g. is holomorphic),then the property also holds for each limit point of the set F. More formally, let X and Y be topological spaces. The set of continuous functions has a natural topology called the compact-open topology. A normal family is a pre-compact subset with respect to this topology. If Y is a metric space, then the compact-open topology is equivalent to the topology of compact convergence, and we obtain a definition which is closer to the classical one: A collection F of continuous functions is called a normal family 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. That is, for every sequence of functions in F, there is a subsequence and a continuous function from X to Y such that the following holds for every compact subset K contained in X: where is the metric of Y. (Wikipedia).

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Topological space | Metric space | Complex analysis | Fundamental normality test | Uniformization theorem | Subsequence | Topology | Riemann sphere | Montel's theorem | Frédéric Marty | Sequence | Compact-open topology | Meromorphic function | Complex plane | Riemann surface | Compact convergence | Mathematics | Function (mathematics) | Riemann mapping theorem | Stereographic projection | Holomorphic function | Compact space | Cauchy's integral theorem | Domain (mathematical analysis) | Relatively compact subspace