Editing Partial isometry (section)

== General definition ==

The concept of partial isometry can be defined in other equivalent ways.  If ''U'' is an isometric map defined on a closed subset  ''H''<sub>1</sub> of a Hilbert space ''H'' then we can define an extension ''W'' of ''U'' to all of ''H'' by the condition that ''W'' be zero on the orthogonal complement of ''H''<sub>1</sub>. Thus a partial isometry is also sometimes defined as a closed partially defined isometric map.

Partial isometries (and projections) can be defined in the more abstract setting of a [[semigroup with involution]]; the definition coincides with the one herein.