Banach limit

We say that the Banach limit is not uniquely determined in this case.

However, there exist sequences for which the values of two Banach limits do not agree.

In other words, a Banach limit extends the usual limits, is shift-invariant and positive.

There are non-convergent sequences which have uniquely determined Banach limits.

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

The most general method for summing a divergent series is non-constructive, and concerns Banach limits.

Banach limit defined on the Banach space that extends the usual limits.

A bounded real sequence is said to be almost convergent to if each Banach limit assigns the same value to the sequence .

In cases where the Cesaro limit does not exist this function can actually be defined as the Banach limit of the indicator functions, which is an extension of this limit.

In mathematical analysis, a Banach limit is a continuous linear functional defined on the Banach space of all bounded complex-valued sequences such that for any sequences and , the following conditions are satisfied: