Proposition 15

Let G be a Hilbert space over K. The mapping $(u, v)\mapsto v\circ u$ of $\mathscr{L}(E; F)\times \mathscr{L}(F; G)$ into $\mathscr{L}(E; G)$ defines by passing to the subspaces a continuous bilinear mapping of norm $\leqslant 1$ of $\mathscr{L}_2(E; F)\times \mathscr{L}_2(F; G)$ into $\mathscr{L}_1(E; G)$.

Let $u\in \mathscr{L}_2(E; F)$ and $v\in \mathscr{L}_2(F; G)$. Let $w\in \mathscr{L}_2(E; G)$. We have $\langle uv|w\rangle =$ Tr($v^*u^*w$) $=\langle v|u^*w\rangle$ whence

by the Cauchy–Schwarz inequality and formula (37) of EVT, V, p 52. The result therefore follows from Prop. 14.