Completeness

Surinder Pal Singh Kainth ()
Additional contact information
Surinder Pal Singh Kainth: Panjab University, Department of Mathematics

Chapter Chapter 4 in A Comprehensive Textbook on Metric Spaces, 2023, pp 89-121 from Springer

Abstract: Abstract Roughly speaking, a metric space X is complete if every sequence in X, which attempts to converge, finds a buddy in X for that purpose. In other words, X is incomplete if it lacks some ‘good’ points. However, it is always possible to extend an incomplete space to a complete one, by appending all such missing ‘good’ points. This chapter starts with a brief introduction to complete metric spaces, followed by its most important application; the Banach Contraction Principle. Then we provide various characterizations of completeness, in terms of Cantor intersection property and totally bounded sets. The completion of a metric space is discussed in a separate section where we establish that the Cauchy completion of $$\mathbb {Q}$$ is isometric to its Dedekind completion. Finally, we present various Banach spaces, including the space of continuous functions, and some results regarding absolute and unconditional convergence.

Date: 2023
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-99-2738-8_4

Ordering information: This item can be ordered from
http://www.springer.com/9789819927388

DOI: 10.1007/978-981-99-2738-8_4

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().