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) (201 ...
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 ...
is said to be ''almost convergent'' to
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^\inf ...
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
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series must ...
. 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