totally bounded

Let A be a subset of a topological vector spaceMathworldPlanetmath X.

A is called totally boundedPlanetmathPlanetmath if , for each neighborhoodMathworldPlanetmathPlanetmath G of 0, there exists a finite subset S of A with A contained in the sumset S+G.

The definition can be restated in the following form when X is a metric space:

A set AX is said to be totally bounded if for every ϵ>0, there exists a finite subset {s1,s2,,sn} of A such that Ak=1nB(sk,ϵ), where B(sk,ϵ) denotes the open ball around sk with radius ϵ.


  • 1 G. Bachman, L. Narici, Functional analysisMathworldPlanetmath, Academic Press, 1966.
  • 2 A. Wilansky, Functional Analysis, Blaisdell Publishing Co., 1964
  • 3 W. Rudin, Functional Analysis, 2nd ed. McGraw-Hill , 1973
Title totally bounded
Canonical name TotallyBounded
Date of creation 2013-03-22 13:09:54
Last modified on 2013-03-22 13:09:54
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 12
Author Mathprof (13753)
Entry type Definition
Classification msc 54E35
Related topic MetricSpace
Related topic BoundedPlanetmathPlanetmathPlanetmathPlanetmath
Related topic Subset