Proposition 3

Let $E_1, ..., E_n$ be hilbertian spaces. Suppose that for $1 \leq i \leq n$ the space $E_i$ is a hilbertian sum of a family $(E_{i,\alpha})_{\alpha \in A(i)}$ of closed vector subspaces. Then $E_1 \otimes_2 ... \otimes_2 E_n$ is a hilbertian sum of the family of subspaces $E_{1,\alpha_1} \hat{\otimes}_2 ... \hat{\otimes}_2 E_{n,\alpha_n}$ with $(\alpha_1, ..., \alpha_n)$ ranging over $A(1) \times ... \times A(n)$.

By formula (6) of V, p. 27, the subspaces $E_{1,\alpha_1} \hat{\otimes}_2 ... \hat{\otimes}_2 E_{n,\alpha_n}$ of $E_1 \hat{\otimes}_2 ... \hat{\otimes}_2 E_n$ are mutually orthogonal. For every integer $i$ between 1 and $n$, the set $\bigcup_{\alpha \in A(i)} E_{i,\alpha}$ is total in $E_i$, and the multilinear mapping $(x_1, ..., x_n) \mapsto x_1 \otimes ... \otimes x_n$ is continuous.

It follows that the union of the subspaces $E_{1,\alpha_1} \hat{\otimes}_2 ... \hat{\otimes}_2 E_{n,\alpha_n}$ is total, hence prop. 3.