Trace-class duality
= Trace-class duality
{title2=$\mathcal S_1(H)^*\cong B(H)$}
The dual of the <trace-class operators>, and also of the normed space of <finite-rank operators> with the <trace norm>, is isometrically the space of <bounded operators>. The pairing is $F_B(T)=\operatorname{Tr}(BT)$. For $R_{x,y}z=\langle z,y\rangle x$, one has $\|R_{x,y}\|_1=\|x\|\|y\|$ and $F_B(R_{x,y})=\langle Bx,y\rangle$, using an <inner product> linear in its first argument. A bounded functional on these <rank-one operators> gives a bounded <sesquilinear form> and hence a unique $B$ by the <Riesz representation theorem>. The <singular value decomposition> proves $\|F_B\|=\|B\|$ and the <triangle inequality> for the <trace norm>.