Proposition 2

For $i = 1, 2$, let $E_i$ and $F_i$ be two Hausdorff prehilbertian spaces and let $u_i \in \mathscr{L}(E_i; F_i)$. The linear mapping $u_1 \otimes u_2$ from $E_1 \otimes_2 E_2$ into $F_1 \otimes_2 F_2$ is continuous and we have

Consider the positive hermitian form on $E_1$ given by

By prop. 1 (V, p. 25), there exists a positive hermitian form $\Phi$ on $E_1 \otimes E_2$ such that

for $x_1, y_1$ in $E_1$ and $x_2, y_2$ in $E_2$. By linearity we have for $z, t$ in $E_1 \otimes E_2$. Since $\Phi$ is positive, we get $\Phi(z, z) \geq 0$, that is, $\|(u_1 \otimes 1).z\|_2$

On the other hand, This completes the proof of prop. 2. Q.E.D.

Let $E_1, ..., E_n$ be prehilbertian spaces ($n \geq 2$). We define the tensor product $E_1 \otimes_2 ... \otimes_2 E_n$ (also denoted by $\bigotimes_{i=1}^n E_i$) by induction, by

Hence, by the definition of scalar product, we have and in particular $^1$ for $x_i, y_i$ in $E_i$ ($1 \leq i \leq n$). If the $E_i$ are Hausdorff, then so is $E_1 \otimes_2 ... \otimes_2 E_n$.

Let $F_1, ..., F_n$ be prehilbertian spaces and $u_i \in \mathscr{L}(E_i; F_i)$ for $1 \leq i \leq n$. By induction on $n$, prop. 2 implies that $u_1 \otimes ... \otimes u_n$ is a continuous linear mapping from $E_1 \otimes_2 ... \otimes_2 E_n$ into $F_1 \otimes_2 ... \otimes_2 F_n$ and that

Let $\sigma \in S_n$ be a permutation of the set $\{1, 2, ..., n\}$. Because of (6), the linear mapping $p_\sigma$ from $E_1 \otimes_2 ... \otimes_2 E_n$ onto $E_{\sigma^{-1}(1)} \otimes_2 ... \otimes_2 E_{\sigma^{-1}(n)}$ characterized by is a prehilbertian space isomorphism ("commutativity of tensor product").

Similarly, consider a partition of $\{1, 2, ..., n\}$ into $m$ consecutive intervals $I_1, ..., I_n$ with $I_k = [a_k, a_{k+1} - 1]$ for $1 \leq k \leq m$. Put

$^1$ Here again we put $\|z\|_2 = \langle z|z \rangle^{1/2}$ for $z$ in $E_1 \otimes_2 ... \otimes_2 E_n$.

The canonical isomorphism from $F_1 \otimes ... \otimes F_m$ onto $E_1 \otimes ... \otimes E_n$ which transforms $\bigotimes_{k=1}^{m} \bigotimes_{i=a_k}^{a_{k+1}-1} x_i$ into $x_1 \otimes ... \otimes x_n$ (A, II, § 3, No. 9) is a prehilbertian space isomorphism (« associativity of the tensor product »).