For a normed space , define to be the set of all equivalent classes of Cauchy sequences obtained by the relation
(1)
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For and , let
(2)
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(3)
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(4)
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Then is a Banach space containing a dense subspace that is isometric with . is called the (Banach) completion of (Kreyszig 1978).
If is a normed algebra, makes into a Banach algebra. Moreover, if is a pre--algebra then equipped with is a -algebra (Murphy 1990).