Example 1

By cor. 1, the space $\ell^2(I) \hat{\otimes}_2 \ell^2(J)$ is canonically isomorphic to $\ell^2(I \times J)$, the tensor product $x \otimes y$ of $x = (x_i)_{i \in I}$ and $y = (y_j)_{j \in J}$ can be identified with the family $(x_i y_j)_{i \in I, j \in J}$. Similarly, by cor. 2, $\ell^2(I) \hat{\otimes}_2 E$ can be identified with $\ell^2_E(I)$, in such a way that we have $(x_i)_{i \in I} \otimes y = (x_i y)_{i \in I}$ for every $y$ in the hilbertian space $E$.