Zero sequence

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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.


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 of null sequences are:

  • ,
  • ,
  • ,
  • ,
  • ,
  • .


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


Web links

Wiktionary: Null sequence  - explanations of meanings, word origins, synonyms, translations