Corollary 2

— Let $E_1$ and $E_2$ be two hilbertian spaces, and $(e_i)_{i \in I}$ an orthonormal basis of $E_1$. Let $(y_i)_{i \in I}$ be a family of elements of $E_2$, such that $\sum_{i \in I} \| y_i \| ^2 < + \infty$. Then the family $(e_i \otimes y_i)_{i \in I}$ is summable in $E_1 \hat{\otimes}_2 E_2$; moreover, every element of $E_1 \otimes E_2$ can be written uniquely in the form $\sum_{i \in I} e_i \otimes y_i$ with $\sum_{i \in I} \| y_i \| ^2 < + \infty$.

Let $F_i$ be the line in $E_1$ generated by the $e_i$ ($i \in I$). Then $E_1$ is the hilbertian sum of the family of subspaces $(F_i)_{i \in I}$. By prop. 3, the space $E_1 \hat{\otimes}_2 E_2$ is the hilbertian sum of the family of subspaces $(F_i \hat{\otimes}_2 E_2)_{i \in I}$, hence cor. 2 follows.