Banach limit


In mathematical analysis, a Banach limit is a continuous linear functional
defined on the Banach space of all bounded complex-valued sequences
such that for all sequences, in, and complex numbers :
  1. ;
  2. if for all, then ;
  3. , where is the shift operator defined by ;
  4. if is a convergent sequence, then.
Hence, is an extension of the continuous functional
where is the complex vector space of all sequences which converge to a limit in.
In other words, a Banach limit extends the usual limits, is linear, shift-invariant and positive.
However, there exist sequences for which the values of two Banach limits do not agree.
We say that the Banach limit is not uniquely determined in this case.
As a consequence of the above properties, a real-valued Banach limit also satisfies:
The existence of Banach limits is usually proved using the Hahn–Banach theorem, or using ultrafilters.
These proofs necessarily use the Axiom of choice.

Almost convergence

There are non-convergent sequences which have a uniquely determined Banach limit.
For example, if,
then is a constant sequence, and
holds.
Thus, for any Banach limit, this sequence has limit.
A bounded sequence with the property, that for every Banach limit the value is the same, is called almost convergent.

Banach spaces

Given a convergent sequence in, the ordinary limit of does not arise from an element of,
if the duality is considered.
The latter means is the continuous dual space of, and consequently,
induces continuous linear functionals on, but not all.
Any Banach limit on is an example of an element of the dual Banach space of
which is not in.
The dual of is known as the ba space, and consists of
all finitely additive measures on the sigma-algebra of all subsets of the natural numbers,
or equivalently,
all Borel measures on the Stone–Čech compactification of the natural numbers.