Banachlimes

In functional analysis , a Banachlimes , named after Stefan Banach , is a functional on the sequence space that is similar to the limit value . ${\displaystyle \ell ^{\infty }}$

definition

In the following denote the left shift ${\displaystyle T}$

${\displaystyle T(x_{n})_{n\in \mathbb {N} }=(x_{n+1})_{n\in \mathbb {N} }}$

and the sequence, which is all ones. ${\displaystyle e=(1,1,1,\ldots )}$

A Banachlimes is a continuous, linear functional that has the following properties: ${\displaystyle \ell \colon \ell ^{\infty }\to \mathbb {R} }$

• ${\displaystyle \ell (e)=1,}$
• for all true ${\displaystyle x\in \ell ^{\infty }}$
  • ${\displaystyle \ell (x)=\ell (Tx),}$
  • if for all , so is ${\displaystyle x_{n}\geq 0}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \ell (x)\geq 0.}$

properties

With the help of Hahn-Banach's theorem one can prove that a Banachlimes exists. However, it is not clearly defined. From the properties required in the definition, it can also be concluded that the classical Limes, which is defined on the space of convergent sequences , continues: ${\displaystyle \ell }$ ${\displaystyle c}$ ${\displaystyle \ell ^{\infty }}$

${\displaystyle \ell (x)=\lim _{n\to \infty }x_{n}}$ For ${\displaystyle x\in c}$

There are non-convergent sequences that have a Banach limit. A simple example of such is

${\displaystyle x=(1,0,1,0,\ldots )}$

Due to the linearity of and the invariance under , the Banach limit of is the same . ${\displaystyle \ell }$ ${\displaystyle T}$ ${\displaystyle x}$ ${\displaystyle 0{,}5}$

The Banach limit value is an example of a functional that does not depend on the shape ${\displaystyle (\ell ^{\infty })'}$

${\displaystyle x\mapsto \sum _{n=1}^{\infty }c_{n}x_{n},\quad c\in \ell ^{1}}$

is.

literature

• Dirk Werner : Functional Analysis . 6th corrected edition. Springer, Berlin 2007, ISBN 978-3-540-72533-6 , pp. 126 .