Completion
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16-7-2021
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Completion
A metric space 
which is not complete has a Cauchy sequence which does not converge. The completion of 
is obtained by adding the limits to the Cauchy sequences.
For example, the rational numbers, with the distance metric, are not complete because there exist Cauchy sequences that do not converge, e.g., 1, 1.4, 1.41, 1.414, ... does not converge because 
is not rational. The completion of the rationals is the real numbers. Note that the completion depends on the metric. For instance, for any prime 
, the rationals have a metric given by the p-adic norm, and then the completion of the rationals is the set of p-adic numbers. Another common example of a completion is the space of L2-functions.
Technically speaking, the completion of 
is the set of Cauchy sequences and 
is contained in this set, isometrically, as the constant sequences.
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