construction of Banach limit using limit along an ultrafilter

Construction of Banach limit using limit along an ultrafilter

The existence of Banach limitMathworldPlanetmath is proved in mathematical analysis usually by Hahn-Banach theoremMathworldPlanetmath. (This proof can be found e.g. in [5], [2] or [4].) Here we will show another approach using limit along a filter. In fact we define it as an -limit of (yn), where (yn) is the Cesàro mean of the sequenceMathworldPlanetmath (xn) and is an arbitrary ultrafilter on .

Theorem 1.

Let F be a free ultrafilter on N. Let (xn) be a bounded ( real sequence. Then the functionalMathworldPlanetmathPlanetmathPlanetmath φ:R


is a Banach limit.


We first observe that φ is defined. Let us denote yn:=x1++xnn. Since (xn) is bounded, the sequence (yn) is bounded as well. Every bounded sequence has a limit along any ultrafilter. This means, that φ(xn)=-limyn exists.

To prove that φ is a Banach limit, we should verify its continuity, positivity, linearity, shift-invariance and to verify that it extends limits.

We first show the shift-invariance. By Sx we denote the sequence xn+1 and we want to show φ(Sx)=φ(x). We observe that x1++xnn-(Sx)1++(Sx)nn=x1++xnn-x2++xn+1n=x1-xn+1n. As the sequence (xn) is bounded, the last expression convergesPlanetmathPlanetmath to 0. Thus φ(x)-φ(Sx)=-limx1-xn+1n=0 and φ(x)=φ(Sx).

The rest of the proof is relatively easy, we only need to use the basic properties of a limit along a filter and of Cesàro mean.

Continuity: x1 |xn|1 |yn|1 |φ(x)|1.

Positivity and linearity follow from positivity and linearity of -limit.

Extends limit: If (xn) is a convergent sequence, then its Cesàro mean (yn) is convergentMathworldPlanetmathPlanetmath to the same limit. ∎


  • 1 B. Balcar and P. Štěpánek, Teorie množin, Academia, Praha, 1986 (Czech).
  • 2 C. Costara and D. Popa, Exercises in functional analysisMathworldPlanetmath, Kluwer, Dordrecht, 2003.
  • 3 K. Hrbacek and T. Jech, Introduction to set theoryMathworldPlanetmath, Marcel Dekker, New York, 1999.
  • 4 T. J. Morisson, Functional analysis: An introduction to Banach spaceMathworldPlanetmath theory, Wiley, 2000.
  • 5 Ch. Swartz, An introduction to functional analysis, Marcel Dekker, New York, 1992.
Title construction of Banach limit using limit along an ultrafilter
Canonical name ConstructionOfBanachLimitUsingLimitAlongAnUltrafilter
Date of creation 2013-03-22 15:32:29
Last modified on 2013-03-22 15:32:29
Owner kompik (10588)
Last modified by kompik (10588)
Numerical id 8
Author kompik (10588)
Entry type Application
Classification msc 03E99
Classification msc 40A05
Related topic BanachLimit