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
ℓ
, and complex numbers
is an extension of the continuous functional
ℓ
is the complex vector space of all sequences which converge to a (usual) 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 (analyst's approach),[1] or using ultrafilters (this approach is more frequent in set-theoretical expositions).
[2] These proofs necessarily use the axiom of choice (so called non-effective proof).
There are non-convergent sequences which have a uniquely determined Banach limit.
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
is the same is called almost convergent.
Given a convergent sequence
, the ordinary limit of
does not arise from an element of
is the continuous dual space (dual Banach space) of
induces continuous linear functionals on
Any Banach limit on
is an example of an element of the dual Banach space of
is known as the ba space, and consists of all (signed) finitely additive measures on the sigma-algebra of all subsets of the natural numbers, or equivalently, all (signed) Borel measures on the Stone–Čech compactification of the natural numbers.