C-20 - Function Spaces

Abstract

Given topological spaces X and Y, C(X, Y) is used to denote the set of all continuous maps from X to Y. It is possible to define various topologies on C(X,Y)—the discrete and indiscrete topologies can always be taken—but for a workable theory of function spaces there should be some relation with the given topologies on X and Y. Every topology weaker than a proper topology is again proper, and every topology stronger than an admissible topology is again admissible. Every proper topology is weaker than every admissible topology, hence there can be only one acceptable topology. Compactness is a very useful property in function spaces, and it is desirable to have characterizations of compactness for subsets of C(X, Y). The Arzèla–Ascoli theorem provides an insight into Montel's theorem on normal families of analytic functions. Variations of the compact-open topology—considering the spaces X and Y—can be used to first specify families K and O of subsets of X and Y respectively. The Stone–Weierstrass theorem characterizes the compactness among the completely regular spaces. There is also a version of this theorem for general spaces.