| One of the central ideas in mathematical analysis is the notion of convergence. In this thesis, the different facets of convergence include point-wise convergence, uniform convergence, and normal convergence. In order to generalize convergence in this way the idea of a metric or generalized measure of distance between two points must be introduced.; A brief discussion of spaces of functions and their related properties form a foundation for the discussion of normal families. Once this foundation has been formed, normal families are formally defined and many of the properties and alternative characterizations of normal families are developed. Beyond these topological conditions for normality, emphasis is placed on a normal family and its underlying metric structure. In particular, metric conditions needed to ensure inheritance of normality of a family to its real and imaginary components, and vice versa, are presented. |
