Let $K$ be a division ring, $E$ a right $K$-vector space and $F$ a left $K$-vector space. Take $u=\sum_{i}x_i\otimes y_i\in E\otimes_KF$. There exists a $u_1\in\text{Hom}_K(E^*,F)$ and $u_2\in\text{Hom}_K(F^*,E)$ s.t. $$u_1(x^*)=\sum_{i}\langle x^*,x_i\rangle y_i\text{ and }u_2(y^*)=\sum_ix_i\langle y_i,y^*\rangle$$ for all $x^*\in E^*$ and $y^*\in F^*$. It can be shown that $\text{rank}(u_1)=\text{rank}(u_2)$.
The author claims that $\text{rank}(u_1)=\text{rank}(u_2)=r$ for some $r\in\mathbf{N}$; he also says that $u_1(E^*)\cong u_2(F^*)^*$ and $u_2(F^*)\cong u_1(E^*)^*$.
Why is the rank finite?