almost convergent

A real sequenceMathworldPlanetmath (xn) is said to be almost convergentMathworldPlanetmathPlanetmath to L if each Banach limit assigns the same value L to the sequence (xn).

Lorentz [4] proved that (xn) is almost convergent to L if and only if


uniformly in n.

The above limit can be rewritten in detail as


Almost convergence is studied in summability theory. It is an example of a summability method which cannot be represented as a matrix method.


  • 1 G. Bennett and N.J. Kalton: Consistency theorems for almost convergence. Trans. Amer. Math. Soc., 198:23–43, 1974.
  • 2 J. Boos: Classical and modern methods in summability. Oxford University Press, New York, 2000.
  • 3 Jeff Connor and K.-G. Grosse-Erdmann: Sequential definitions of continuity for real functions. Rocky Mt. J. Math., 33(1):93–121, 2003.
  • 4 G. G. Lorentz: A contribution to the theory of divergent sequencesMathworldPlanetmath. Acta Math., 80:167–190, 1948.
Title almost convergent
Canonical name AlmostConvergent
Date of creation 2013-03-22 15:29:51
Last modified on 2013-03-22 15:29:51
Owner kompik (10588)
Last modified by kompik (10588)
Numerical id 12
Author kompik (10588)
Entry type Definition
Classification msc 40A05
Classification msc 40C99
Related topic Banachlimit
Defines almost convergent