QUASI-UNIFORM AND UNIFORM CONVERGENCE OF RIEMANN AND RIEMANN-TYPE INTEGRABLE FUNCTIONS WITH VALUES IN A BANACH SPACE

Abstract

In this article, we study quasi-uniform and uniform convergenceof nets and sequences of different types of functions defined on a topological space, in particular on a closed bounded interval of $\mathbb{R}$, with values in a metric space and in some cases in a Banach space. We show that boundedness and continuity are inherited to the quasi-uniform limit, and integrability is inherited to the uniform limit of a net of functions. Given a sequence of functions, we construct functions with values in a sequence space and consequently we infer some important properties of such functions. Finally, we study convergence of partially equi-regulated* nets of functions which is shown to be a generalized notion of exhaustiveness.