Exercise 11, § 4

Let $E$ and $F$ be two hilbertian spaces. Show that for every element $u$ in $E \otimes_2 F$, there exists an orthonormal sequence $(e_n)$ in $E$, an orthonormal sequence $(f_n)$ in $F$ and a sequence $(\lambda_n)$ of numbers $\geq 0$ such that $\sum \lambda_n^2 < +\infty$ and that $u = \sum \lambda_n e_n \otimes f_n$; then $\|u\|_2^2 = \sum \lambda_n^2$ (cf. V, p. 55, th. 2 and p. 53, th. 1).