Function Spaces

Simple convergence (= pointwise convergence) and uniform convergence known from Analysis are studied first in the realm of classical General Topology. Also continuous convergence introduced by H. Hahn [56] is considered in this context. Since pointwise convergence can be described by means of the product topology, which was first observed by A. Tychonoff [143], uniform spaces are needed for uniform convergence (the uniformity of uniform convergence was first explicitly defined by J.W. Tukey [141]). In order to study continuous convergence in the realm of topological spaces, the restriction to locally compact Hausdorff spaces is necessary (cf. theorem 6.1.31.), i.e. for infinite-dimensional analysis continuous convergence cannot be described in this framework. Since the complex plane is a locally compact Hausdorff space, in classical Function Theory continuous convergence is available and according to C. Carathéodory [25], it is often useful to substitute uniform convergence on compacta by continuous convergence. By the way, if we consider locally compact Hausdorff spaces, then the topology describing continuous convergence is the compact-open topology, introduced and studied first by R.H. Fox [45] and R. Arens [4]. By the introduction of this book, the reason why topological spaces are not sufficient for studying continuous convergence can also be formulated as follows: Top is not cartesian closed.