Banach limit

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In mathematical analysis, a Banach limit is a continuous linear functional $\phi: \ell_\infty \to \mathbb{R}$ defined on the Banach space $\ell_\infty$ of all bounded real-valued sequences such that for any real-valued sequences $x = (x n)$ and $y = (y n)$ , the following conditions are satisfied:

$φ(c . x + d . y) = c .φ(x) + d .φ(y)$ (linearity);
if $x\geq 0$ , then $\phi(x)\geq 0$ ;
$φ(x) = φ(S x)$ , where $S$ is the shift operator defined by $(S x) n = x n + 1$ .
If $x$ is a convergent sequence, then $\phi(x)=\lim x$ .

In other words, a Banach limit extends the usual limits, is 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.

The existence of Banach limits is usually proved using the Hahn-Banach theorem (analyst's approach) or using ultrafilters (this approach is more frequent in set-theoretical expositions). It is worth mentioning, that these proofs use Axiom of choice (so called non-effective proof).

[edit] Almost convergence

Some sequences have uniquely determined Banach limits. For example, if $x=(1,0,1,0,\ldots)$ , then $x+S(x)=(1,1,1,\ldots)$ is a constant sequence, and it holds $2φ(x) = φ(x) + φ(S x) = 1$ . Thus for any Banach limit this sequence has limit $\frac 12$ .