Proposition 17

Let u be an element of $E \otimes_K F$ and $M \subset E$ and $N \subset F$ its associated subspaces. For every expression $u = \sum_{i=1}^s x_i \otimes y_i$ of $u$, where $x_i \in E$ and $y_i \in F$ for $1 \leq i \leq s$, the subspace $M$ (resp. $N$) is contained in the subspace of $E$ (resp. $F$) generated by the $x_i$ (resp. the $y_i$). Moreover, the following properties are equivalent:

(a) The integer $s$ is equal to the rank of $u$. (b) The family $(x_i)_{1 \leq i \leq s}$ is a basis of $M$. (c) The family $(y_i)_{1 \leq i \leq s}$ is a basis of $N$. (d) The families $(x_i)_{1 \leq i \leq s}$ and $(y_i)_{1 \leq i \leq s}$ are both free.

By (33) (resp. (34)) each element of $N = u_1(E^*)$ (resp. $M = u_2(F^*)$) is a linear combination of the $y_i$ (resp. the $x_i$); whence the first assertion. If $s = r$, the subspace generated by the $x_i$ (resp. $y_i$) with dimension $\leq \dim M$ (resp. $\leq \dim N$) and containing $M$ (resp. $N$) is identical with it and hence (a) implies (b) and (c) and a fortiori (d). Conversely, each of conditions (b) and (c) implies (a) by definition of $\mathrm{rg}(u)$. Finally if (d) holds, there exists a family $(x_i^*)_{1 \leq i \leq s}$ of elements of $E^*$ such that $\langle x_i, x_j^* \rangle = \delta_{ij}$ (no. 5, Corollary 1 to Theorem 7) and hence it follows from (33) that $(y_i)$ is a basis of $N$, which completes the proof.