# Banach limit

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 complex-valued sequences such that for any sequences $x=(x_n)$ and $y=(y_n)$, the following conditions are satisfied:

1. $\phi(\alpha x+\beta y)=\alpha\phi(x)+\beta \phi(y)$ (linearity);
2. if $x_n\geq 0$ for all $n\ge1$, then $\phi(x)\geq 0$;
3. $\phi(x)=\phi(Sx)$, where $S$ is the shift operator defined by $(Sx)_n=x_{n+1}$.
4. If $x$ is a convergent sequence, then $\phi(x)=\lim x$.

Hence, $\phi$ is an extension of the continuous functional $\lim x:c\mapsto \mathbb C.$

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. However, as a consequence of the above properties, a Banach limit also satisfies:

$\liminf_ {n\to\infty} x_n\le\phi(x) \le \limsup_{n\to\infty}x_n$

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). These proofs necessarily use the Axiom of choice (so called non-effective proof).

## Almost convergence

There are non-convergent sequences which 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 $2\phi(x)=\phi(x)+\phi(Sx)=1$ holds. Thus for any Banach limit this sequence has limit $\frac 12$.

A sequence $x$ with the property, that for every Banach limit $\phi$ the value $\phi(x)$ is the same, is called almost convergent.

## Ba spaces

Given a sequence in c, the ordinary limit of the sequence does not arise from an element of $\ell^1$. Thus the Banach limit on $\ell^\infty$ is an example of an element of the continuous dual space of $\ell^\infty$ which is not in $\ell^1$. The dual of $\ell^\infty$ 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, (signed) Borel measures on the Stone–Čech compactification of the natural numbers.