Corollary 2

Let F be a Hilbert space. Let $u\in \mathscr{L}(E; F)$ be a Hilbert-Schmidt mapping. Then $u$ is a compact linear mapping.

Let $(j,|u|)$ be the polar decomposition of $u($I, p. 140, def. 4). By definition (EVT, V, p. 50, def. 9) the endomorphism $u^*u$ is of finite trace,$\surd$ hence is compact (corollary 1, a)); the same is true of $|u|=u^*u$ (prop. 6 of III, p. 91, b)) and of $u=j|u|$ (prop. 3 of III, p. 5).