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 analysisusually by Hahn-Banach theorem. (This proof can be found e.g. in[5], [2] or [4].) Here wewill show another approach using limit along a filter. In fact wedefine it as an $ℱ$-limit of $(y_n)$, where $(y_n)$ is theCesàro mean of the sequence $(x_n)$ and $ℱ$ is an arbitraryultrafilter on $\mathbb{N}$.

Theorem 1.

Let $\mathrm{F}$ be a free ultrafilter on $\mathrm{N}$. Let $\mathrm{(}{x}_n\mathrm{)}$ be a bounded (http://planetmath.org/Bounded) realsequence. Then the functional $\phi \mathrm{:}{\mathrm{\ell }}_{\mathrm{\infty }}\mathrm{\to }\mathrm{R}$

$$\phi (x_n)=ℱ\mathit{\text{-}}lim\frac{x_1+\mathrm{\dots }+x_n}{n}$$

is a Banach limit.

Proof.

We first observe that $\phi$ is defined. Let us denote$y_n:=\frac{x_1+\mathrm{\dots }+x_n}{n}$. Since $(x_n)$ is bounded, thesequence $(y_n)$ is bounded as well. Every bounded sequence has alimit along any ultrafilter. This means, that $\phi (x_n)=ℱ\text{-}lim{y}_n$exists.

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

We first show the shift-invariance. By $Sx$ we denote the sequence$x_{n+1}$ and we want to show $\phi (Sx)=\phi (x)$. We observe that$\frac{x_1+\mathrm{\dots }+x_n}{n}-\frac{{(Sx)}_1+\mathrm{\dots }+{(Sx)}_n}{n}=\frac{x_1+\mathrm{\dots }+x_n}{n}-\frac{x_2+\mathrm{\dots }+x_{n+1}}{n}=\frac{x_1-x_{n+1}}{n}$. As the sequence $(x_n)$ is bounded, the lastexpression converges to 0. Thus $\phi (x)-\phi (Sx)=ℱ\text{-}lim\frac{x_1-x_{n+1}}{n}=0$ and $\phi (x)=\phi (Sx)$.

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

Continuity: $\parallel x\parallel \le 1$ $\Rightarrow$ $|x_n|\le 1$ $\Rightarrow$$|y_n|\le 1$ $\Rightarrow$ $|\phi (x)|\le 1$.

Positivity and linearity follow from positivity and linearity of$ℱ$-limit.

Extends limit: If $(x_n)$ is a convergent sequence, then itsCesàro mean $(y_n)$ is convergent to the same limit.∎