Definition 1

Let $E_1, ..., E_n$ be hilbertian spaces. The completion of the Hausdorff prehilbertian space $E_1 \hat{\otimes}_2 ... \hat{\otimes}_2 E_n$ is called the hilbertian tensor product of the $E_i$ and is denoted by $E_1 \hat{\otimes}_2 ... \hat{\otimes}_2 E_n$ (or $\hat{\otimes}_{2 \atop 1 \leq i \leq n} E_i$).

Let $F_1, ..., F_n$ be hilbertian spaces and $u_i \in \mathscr{L}(E_i, F_i)$ for $1 \leq i \leq n$. The continuous linear mapping $u_1 \otimes ... \otimes u_n$ then extends to a continuous linear mapping $u_1 \hat{\otimes}_2 ... \hat{\otimes}_2 u_n$ from $E_1 \hat{\otimes}_2 ... \hat{\otimes}_2 E_n$ into $F_1 \hat{\otimes}_2 ... \hat{\otimes}_2 F_n$. We have

by formula (8) of V, p. 27. Moreover, if $1_E$ denotes the identity mapping of any hilbertian space $E$, we have

Finally, if $G_1, ..., G_n$ are hilbertian spaces and $v_i \in \mathscr{L}(F_i; G_i)$ for $1 \leq i \leq n$, we get

We leave to the reader the task of formulating the « commutativity » and the « associativity » of the hilbertian tensor product, in analogy with what has been said above for prehilbertian spaces.