Zero sequence
In mathematics , a zero sequence is a sequence (mostly of real numbers ) that converges (approaches) towards 0 . Each convergent sequence can be represented as the sum of a constant number (namely its limit value) and a zero sequence.
For example, the sequence is a zero sequence of real numbers.
definition
Be the field of real or complex numbers. A sequence is called a null sequence if
applies. The set of all zero sequences forms the sequence space , which becomes a Banach space with the supremum norm .
Examples
Examples of null sequences are:
- ,
- ,
- ,
- ,
- ,
- .
generalization
Let be a metrizable topological group , i. H. a group that is equipped with a metric in such a way that the group connection and the formation of the inverse are continuous (e.g. the additive group in a weighted field or normalized vector space ).
A sequence in is called a zero sequence if and only if it converges to the neutral element.
The property of a result to be null sequence, of course, depends on the metric from: The above mentioned as an example result in a null sequence with respect to the usual amount of metric, but it diverges even with respect to the 2-adic amount to .
A sequence in a standardized vector space is a null sequence with respect to the metric induced by the norm if and only if the sequence of the norms is a null sequence in .
See also
swell
- Dirk Werner : Functional Analysis. Sixth, corrected edition. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , page 8