Compactness in Metric Spaces

Radu Precup
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Radu Precup: Babeş-Bolyai University, Department of Applied Mathematics

Chapter Chapter 1 in Methods in Nonlinear Integral Equations, 2002, pp 13-23 from Springer

Abstract: 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

Keywords: Banach Space; Compact Subset; Open Ball; Cauchy Sequence; Convergent Subsequence (search for similar items in EconPapers)
Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-015-9986-3_2

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DOI: 10.1007/978-94-015-9986-3_2

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