A metric space X is completePlanetmathPlanetmathPlanetmathPlanetmath if every Cauchy sequenceMathworldPlanetmathPlanetmath ( in X is a convergent sequenceMathworldPlanetmath.



Cauchy sequence

  • The space of rational numbers is not complete: the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592 consisting of finite decimals converging to π is a Cauchy sequence in that does not converge in .

  • The space of real numbers is complete, as it is the completion of with respect to the standard metric (other completions, such as the p-adic numbers, are also possible). More generally, the completion of any metric space is a complete metric space.

  • Every Banach spaceMathworldPlanetmath is complete. For example, the Lp–space of p-integrable functions is a complete metric space if p1.

Title complete
Canonical name Complete
Date of creation 2013-03-22 11:55:11
Last modified on 2013-03-22 11:55:11
Owner djao (24)
Last modified by djao (24)
Numerical id 12
Author djao (24)
Entry type Definition
Classification msc 54E50