# construction of Banach limit using limit along an ultrafilter

## Construction of Banach limit using limit along an ultrafilter

The existence of Banach limit is proved in mathematical analysis usually by Hahn-Banach theorem. (This proof can be found e.g. in [5], [2] or [4].) Here we will show another approach using limit along a filter. In fact we define it as an $\mathcal{F}$-limit of $(y_{n})$, where $(y_{n})$ is the Cesàro mean of the sequence $(x_{n})$ and $\mathcal{F}$ is an arbitrary ultrafilter on $\mathbb{N}$.

###### Theorem 1.

Let $\mathcal{F}$ be a free ultrafilter on $\mathbb{N}$. Let $(x_{n})$ be a bounded (http://planetmath.org/Bounded) real sequence. Then the functional $\varphi:\ell_{\infty}\to\mathbbmss{R}$

 $\varphi(x_{n})=\operatorname{\mathcal{F}\text{-}\lim}\frac{x_{1}+\ldots+x_{n}}% {n}$

is a Banach limit.

###### Proof.

We first observe that $\varphi$ is defined. Let us denote $y_{n}:=\frac{x_{1}+\ldots+x_{n}}{n}$. Since $(x_{n})$ is bounded, the sequence $(y_{n})$ is bounded as well. Every bounded sequence has a limit along any ultrafilter. This means, that $\varphi(x_{n})=\operatorname{\mathcal{F}\text{-}\lim}y_{n}$ exists.

To prove that $\varphi$ is a Banach limit, we should verify its continuity, positivity, linearity, shift-invariance and to verify that it extends limits.

We first show the shift-invariance. By $Sx$ we denote the sequence $x_{n+1}$ and we want to show $\varphi(Sx)=\varphi(x)$. We observe that $\frac{x_{1}+\ldots+x_{n}}{n}-\frac{(Sx)_{1}+\ldots+(Sx)_{n}}{n}=\frac{x_{1}+% \ldots+x_{n}}{n}-\frac{x_{2}+\ldots+x_{n+1}}{n}=\frac{x_{1}-x_{n+1}}{n}$. As the sequence $(x_{n})$ is bounded, the last expression converges to 0. Thus $\varphi(x)-\varphi(Sx)=\operatorname{\mathcal{F}\text{-}\lim}\frac{x_{1}-x_{n+% 1}}{n}=0$ and $\varphi(x)=\varphi(Sx)$.

The rest of the proof is relatively easy, we only need to use the basic properties of a limit along a filter and of Cesàro mean.

Continuity: $\lVert x\rVert\leq 1$ $\Rightarrow$ $|x_{n}|\leq 1$ $\Rightarrow$ $|y_{n}|\leq 1$ $\Rightarrow$ $|\varphi(x)|\leq 1$.

Positivity and linearity follow from positivity and linearity of $\mathcal{F}$-limit.

Extends limit: If $(x_{n})$ is a convergent sequence, then its Cesàro mean $(y_{n})$ is convergent to the same limit. ∎

## References

• 1 B. Balcar and P. Štěpánek, Teorie množin, Academia, Praha, 1986 (Czech).
• 2 C. Costara and D. Popa, Exercises in functional analysis, Kluwer, Dordrecht, 2003.
• 3 K. Hrbacek and T. Jech, Introduction to set theory, Marcel Dekker, New York, 1999.
• 4 T. J. Morisson, Functional analysis: An introduction to Banach space theory, Wiley, 2000.
• 5 Ch. Swartz, An introduction to functional analysis, Marcel Dekker, New York, 1992.
Title construction of Banach limit using limit along an ultrafilter ConstructionOfBanachLimitUsingLimitAlongAnUltrafilter 2013-03-22 15:32:29 2013-03-22 15:32:29 kompik (10588) kompik (10588) 8 kompik (10588) Application msc 03E99 msc 40A05 BanachLimit