Remark

Let $E_1, ..., E_n$ be Hausdorff prehilbertian spaces, and $\hat{E}_1, ..., \hat{E}_n$ their respective completions. Then $E_1 \otimes_2 ... \otimes_2 E_n$ is a prehilbertian subspace of $\hat{E}_1 \otimes_2 ... \otimes_2 \hat{E}_n$. Since the mapping $(x_1, ..., x_n) \mapsto x_1 \otimes ... \otimes x_n$ from $\hat{E}_1 \times ... \times \hat{E}_n$ into $\hat{E}_1 \otimes_2 ... \otimes_2 \hat{E}_n$ is continuous, $E_1 \otimes_2 ... \otimes_2 E_n$ is dense in $\hat{E}_1 \otimes_2 ... \otimes_2 \hat{E}_n$. A fortiori the completion of $E_1 \otimes_2 ... \otimes_2 E_n$ is precisely the hilbertian space $\hat{E}_1 \hat{\otimes}_2 ... \hat{\otimes}_2 \hat{E}_n$. This completion is sometimes simply written as $E_1 \hat{\otimes}_2 ... \hat{\otimes}_2 E_n$ (or $\hat{\otimes}_{2 \atop 1 \leq i \leq n} E_i$).