Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 106 / 4 / b

The estimateshows $\Vert {\mu }_{x,y}\Vert \le \Vert x\Vert \Vert y\Vert$. For nonzero $x,y$, the rank-one operatorhas norm one and attains equality. The zero cases are immediate.

Embed the operator unit ball intoby $T\mapsto (⟨Tx,y⟩{)}_{x,y}$. The product is compact by Tychonoff theorem. A pointwise limit of these coordinates is a bilinear form $\theta$ satisfyingBy the stated representation theorem for bounded bilinear forms, $\theta (x,y)=⟨Tx,y⟩$ for a unique operator with $\Vert T\Vert \le 1$. The image is therefore closed and compact. Its product topology is precisely the weak operator topology $\sigma (\mathcal{B}(H),M)$.

The linear span $Z=\mathrm{span}M$ separates operators, and the preceding compactness lets part (a) identify $\mathcal{B}(H)$ isometrically with $Z^{\ast }$. Thus $\mathcal{B}(H)$ is a dual Banach space. This is the operator predual from matrix coefficients.