A bounded real sequence is said to be almost convergent if each Banach limit.

is almost convergent if and only if

.

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.

References

• 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.

• .

• Specific