Editing Banach space (section)

====Completions====

Every normed space can be [[isometry|isometrically]] embedded onto a dense vector subspace of a Banach space, where this Banach space is called a ''[[Completion (metric space)|completion]]'' of the normed space. This Hausdorff completion is unique up to [[Isometry|isometric]] isomorphism.

More precisely, for every normed space <math>X,</math> there exists a Banach space <math>Y</math> and a mapping <math>T : X \to Y</math> such that <math>T</math> is an [[Isometry|isometric mapping]] and <math>T(X)</math> is dense in <math>Y.</math> If <math>Z</math> is another Banach space such that there is an isometric isomorphism from <math>X</math> onto a dense subset of <math>Z,</math> then <math>Z</math> is isometrically isomorphic to <math>Y.</math>
The Banach space <math>Y</math> is the Hausdorff ''[[Complete metric space#Completion|completion]]'' of the normed space <math>X.</math> The underlying metric space for <math>Y</math> is the same as the metric completion of <math>X,</math> with the vector space operations extended from <math>X</math> to <math>Y.</math> The completion of <math>X</math> is sometimes denoted by <math>\widehat{X}.</math>