Consider the set of all convergent complex-valued sequences . The limit operation is a linear functional on , by the usual limit laws. A Banach limit is, loosely speaking, any linear functional that generalizes to apply to non-convergent sequences as well. The formal definition follows:
if (That is, extends .)
, where is the shift operator defined by . (Shift invariance)
If for all , then . (Positivity)
Like the limit superior and limit inferior, the Banach limit can be applied for situations where one wants to algebraically manipulate limit equations or inequalities, even when it is not assured beforehand that the limits in question exist (in the classical sense).
1 Some consequences of the definition
The positivity condition ensures that the Banach limit of a real-valued sequence is real-valued, and that limits can be compared: if , then . In particular, given a real-valued sequence , by comparison with the sequences and , it follows that .
The shift invariance allows any finite number of terms of the sequence to be neglected when taking the Banach limit, as is possible with the classical limit.
On the other hand, can never be multiplicative, meaning that fails. For a counter-example, set ; then we would have , so , but .
2 Other definitions
In some definitions of the Banach limit, condition (i) is replaced by the seemingly weaker condition that — the Banach limit of a constant sequence is that constant. In fact, the latter condition together with shift invarance implies condition (i).
If we restrict to real-valued sequences, condition (ii) is clearly redundant, in view of the other conditions.
|Date of creation||2013-03-22 15:23:00|
|Last modified on||2013-03-22 15:23:00|
|Last modified by||stevecheng (10074)|