Compactness in Metric Spaces

Abstract

In this chapter we first define the notions of a compact metric space and of a relatively compact subset of a metric space. Then we state and prove Hausdorff’s theorem of the characterization of the relatively compact subsets of a complete metric space in terms of finite and relatively compact ε--nets. Furthermore, we prove the Ascoli-Arzèla and Fréchet-Kolmogorov theorems of characterization of the relatively compact subsets of C (K; R ⁿ) and L ^p (Ω; R ⁿ), respectively. Here K is a compact metric space, Ω ⊂ R ^N is a bounded open set and 1 ≤ p < ∞.