zero sequence

Let a field k be equipped with a rank one valuation |.|.  A sequence

a1,a2, (1)

of elements of k is called a zero sequence or a null sequence, if  limnan=0  in the metric induced by |.|.

If k together with the metric induced by its valuationMathworldPlanetmath |.| is a complete ultrametric field, it’s clear that its sequence (1) has a limit (in k) as soon as the sequence


is a zero sequence.

If k is not completePlanetmathPlanetmathPlanetmathPlanetmath with respect to its valuation |.|, its completion ( can be made as follows.  The Cauchy sequencesPlanetmathPlanetmath (1) form an integral domainMathworldPlanetmath D when the operations “+” and “” are defined componentwise.  The subset P of D formed by the zero sequences is a maximal idealMathworldPlanetmathPlanetmath, whence the quotient ringMathworldPlanetmath D/P is a field K.  Moreover, k may be isomorphically embedded into K and the valuation |.| may be uniquely extended to a valuation of K.  The field K then is complete with respect to |.| and k is dense in K.

Title zero sequence
Canonical name ZeroSequence
Date of creation 2015-07-10 21:03:45
Last modified on 2015-07-10 21:03:45
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Definition
Classification msc 40A05
Synonym null sequence