Metrizability of Compact Sets in the Class $\mathfrak {G}$

Abstract

This chapter introduces (after Cascales and Orihuela) a large class of locally convex spaces under the name the class $\mathfrak{G}$. The class $\mathfrak{G}$ contains among others all (LM)-spaces (hence (LF)-spaces), and dual metric spaces (hence (DF)-spaces), spaces of distributions D′(Ω) and spaces A(Ω) of real analytic functions on open Ω⊂ℝⁿ. We show (following Cascales and Orihuela) that every precompact set in an lcs in the class $\mathfrak{G}$ is metrizable. This general result covers many already known theorems for (DF)-spaces, (LF)-spaces and dual metric spaces.