Exercise 11, § 4

Let $E$ and $F$ be two Hilbert spaces. Now prove that, for every element $u$ of $E \otimes_2 F$, 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 $u = \sum \lambda_n e_n \otimes f_n$; one then has $\|u\|_2^2 = \sum \lambda_n^2$ (cf. V, p. 54, th. 2 et p. 52, th. 1).