On realcompact topological vector spaces

This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vector spaces appearing in Functional Analysis, including Fréchet spaces, ( LF )-spaces, and their duals, ( DF )-spaces and spaces of continuous real-valued functions C ( X ) on a completely regular Hausdorff space X . Especially ( LF )-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis.