Almost convergent sequence

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A bounded real sequence is said to be almost convergent to if each Banach limit assigns the same value to the sequence .

Lorentz proved that is almost convergent if and only if

uniformly in .

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.
  • J. Boos: "Classical and modern methods in summability." Oxford University Press, New York, 2000.
  • J. Connor and K.-G. Grosse-Erdmann: "Sequential definitions of continuity for real functions." Rocky Mt. J. Math., 33(1):93--121, 2003.
  • G.G. Lorentz: "A contribution to the theory of divergent sequences." Acta Math., 80:167--190, 1948.
  • Hardy, G. H. (1949), Divergent Series, Oxford: Clarendon Press .

This article incorporates material from Almost convergent on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

  1. ^ Hardy,p.52