Integration of Continuous Functions

This chapter defines the Cauchy integral of a uniformly continuous function taking values in a Neumann space. It reviews the Neumann spaces and continuous mappings. The chapter defines separated semi-normed spaces, convergent sequences and Cauchy sequences in a separated semi-normed space. It describes the bounded subsets of a separated semi-normed space, several types of sequentially complete spaces, and filtering families of semi-norms. The use of filtering families simplifies certain statements by replacing the supremum over a finite number of semi-norms with a single semi-norm. Every continuous mapping from a subset of a separated semi-normed space into a separated semi-normed space is sequentially continuous. The chapter provides some growth properties of the real-valued integral. It shows that the order of integration of successive variables is indifferent.