Almost uniform convergence versus pointwise convergence

Abstract

In many an example of a function space whose topology is the topology of almost uniform convergence it is observed that the same topology is obtained in a natural way by considering pointwise convergence of extensions of the functions on a larger domain [l; 2].This paper displays necessary conditions and sufficient conditions for the above situation to occur.Consider a linear space G(5, F) of functions with domain 5 and range in a real or complex locally convex linear topological space £.Assume that there are sufficient functions in G(5, £) to distinguish between points of 5. Let Sß denote the closure of the image of 5 in the cartesian product space X{g(5): g£G(5, £)}.Theorems 4.1 and 4.2 of reference [2] give the following theorem.

Locations

• Proceedings of the American Mathematical Society - View - PDF