Function Spaces

Abstract

In this chapter, we shall learn various methods of introducing topologies on a collection of functions of a space X into another space Y. The reader might have studied the notions of pointwise convergence and uniform convergence in Analysis. Section 11.1 concerns with generalizations of these notions in the realm of topological spaces and two topologies for a family of functions $X\rightarrow Y$, one describing the pointwise convergence and the other describing the uniform convergence. The next section is devoted to the “compact-open topology”, which is predominantly preferred for function spaces. In Sect. 11.3, we confine ourselves to families of functions of a topological space into a metric space. We shall discuss here “the topology of compact convergence” and prove a theorem for compactness of a function space in this topology, known as Arzela–Ascoli theorem.