A bounded real sequence
is said to be almost convergent
if each Banach limit
the same value
to the sequence
Lorentz proved that 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.
- 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.