Lemma 8

The mapping $u\mapsto u^*$ of $\mathscr{L}(E; F)$ into $\mathscr{L}(F; E)$ defines by passing to the subspaces an isometric linear mapping of $\mathscr{L}_1(E; F)$ into $\mathscr{L}_1(F; E)$.

Let $u\in \mathscr{L}_1(E; F)$. Since $u$ is a Hilbert–Schmidt mapping, the same is true of $u^*$ (EVT, V, p. 54). The mapping $v\mapsto v^*$ is a bijection of the set of $v\in \mathscr{L}_2(E; F)$ such that $\|v\|\leqslant 1$ onto the set of $w\in \mathscr{L}_2(F; E)$ such that $\|w\|\leqslant 1$; for every $v\in \mathscr{L}_2(E; F)$ with $\|v\|\leqslant 1$, one has $\langle v|u\rangle =\langle u^*|v^*\rangle$ (EVT, V, p. 54, formula (42)), whence the result by Prop. 14.