Banach limit

In the functional analysis is a Banachlimes, named after Stefan Banach, a limit similar to the functional on the sequence space.

Definition

In the following denote the left Shift

And the sequence is all ones.

A Banachlimes is a continuous, linear functional, which has the following properties:

• Applies to all
• If for all, so is

Properties

With the help of the theorem of Hahn- Banach can be proved that a Banachlimes exists. However, it is not uniquely determined. From the information required in the definition of properties, also it can be concluded that the classical limit, which is defined on the space of convergent sequences, after continuing:

There are non- convergent sequences which have a Banachgrenzwert. A simple example of such is

Due to the linearity of and the invariance under the Banachgrenzwert of the same.

The Banachgrenzwert is of an example of a functional, not from the shape

Is.