Editing Hilbert–Schmidt operator (section)

== Space of Hilbert–Schmidt operators ==

The product of two Hilbert–Schmidt operators has finite [[trace class|trace-class norm]]; therefore, if ''A'' and ''B'' are two Hilbert–Schmidt operators, the '''Hilbert–Schmidt inner product''' can be defined as

<math display="block">\langle A, B \rangle_\text{HS} = \operatorname{Tr}(B^* A) = \sum_i \langle Ae_i, Be_i \rangle.</math>

The Hilbert–Schmidt operators form a two-sided [[ideal (ring theory)|*-ideal]] in the [[Banach algebra]] of bounded operators on {{math|''H''}}. 
They also form a Hilbert space, denoted by {{math|''B''<sub>HS</sub>(''H'')}} or {{math|''B''<sub>2</sub>(''H'')}}, which can be shown to be [[Natural transformation|naturally]] isometrically isomorphic to the [[tensor product of Hilbert spaces]]

<math display="block">H^* \otimes H,</math>

where {{math|''H''<sup>∗</sup>}} is the [[dual space]] of {{math|''H''}}. 
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space).{{sfn | Conway | 1990 | p=268}} 
The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with the Hilbert–Schmidt norm).{{sfn | Conway | 1990 | p=268}} 

The set of Hilbert–Schmidt operators is closed in the [[norm topology]] if, and only if, {{math|''H''}} is finite-dimensional.