A bounded

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

(x_n)

is said to be ''almost convergent'' to

L

if each

Banach limit In mathematical analysis, a Banach limit is a continuous linear functional \phi: \ell^\infty \to \mathbb defined on the Banach space \ell^\infty of all bounded complex-valued sequences such that for all sequences x = (x_n), y = (y_n) in \ell^\in ...

assigns the same value

L

to the sequence

(x_n)

. Lorentz proved that

(x_n)

is almost convergent if and only if :

\lim\limits_ \fracp=L

uniformly in

n

. The above limit can be rewritten in detail as :

\forall \varepsilon>0 : \exists p_0 : \forall p>p_0 : \forall n : \left, \fracp-L\<\varepsilon.

Almost convergence is studied in summability theory. It is an example of a summability method which cannot be represented as a matrix method.Hardy,p.52

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 {{PlanetMath attribution, urlname=almostconvergent, title=Almost convergent Convergence (mathematics) Sequences and series