Editing Trace class (section)

=== As the predual of bounded operators ===

Recall that the dual of <math>\ell^1(\N)</math> is <math>\ell^{\infty}(\N).</math> In the present context, the dual of trace-class operators <math>B_1</math> is the bounded operators <math>B(H).</math> More precisely, the set <math>B_1</math> is a two-sided [[Ideal (ring theory)|ideal]] in <math>B(H).</math> So given any operator <math>T \in B(H),</math> we may define a [[Continuous linear operator|continuous]] [[linear functional]] <math>\varphi_T</math> on <math>B_1</math> by <math>\varphi_T(A) = \operatorname{Tr} (AT).</math> This correspondence between bounded linear operators and elements <math>\varphi_T</math> of the [[dual space]] of <math>B_1</math> is an [[isometric isomorphism]]. It follows that <math>B(H)</math> {{em|is}} the dual space of <math>B_1.</math> This can be used to define the [[Weak-star operator topology|weak-* topology]] on <math>B(H).</math>