Lemma 9

Let $u\in \mathscr{L}_1(E; F)$. We have

$\|u\|_1=$ sup$_{v\in\mathscr{L}_c(F;E)}|b(u, v)|$.

$\|v\|\leqslant 1$

Since every Hilbert-Schmidt mapping is compact (cor. 2 of IV, p. 165), we have

$\|u\|_1\leqslant$ sup$_{v\in\|\mathscr{L}v\|^c(E;F)\leqslant 1}|$Tr($vu$)$|=$ sup$_{v\in\mathscr{L}\|v^c\|(F;E)\leqslant 1}|b(u, v)|$

by prop. 14.

Let $v\in \mathscr{L}^c(E; F)$ be of norm $\leqslant$ 1. Since $\mathscr{L}_2(E; F)$ is dense in $\mathscr{L}^c(E; F)$, there exists a sequence $(v_n)_{n\in\mathbf{N}}$ in $\mathscr{L}_2(E; F)$ which converges to $v$ in $\mathscr{L}(E; F)$. The sequence $(b(u, v_n))_{n\in\mathbf{N}}$ converges to $b(u, v)$; since $|b(u, v_n)|$ = $|$Tr($v_nu$)$|$ = $|\langle v_n^*|u\rangle |\leqslant \|u\|_1$ (prop. 14) for every $n\in \mathbf{N}$, it follows that $|b(u, v)|\leqslant \|u\|_1$.

Let us denote by $\theta$ the continuous linear mapping

such that $\theta (u)(v) =b(u, v)$.