Definition 7

Let A be a field (resp. a field endowed with an antiautomorphism J), E a left vector space over A, F a right (resp. left) vector space over A, and $\Phi$ a bilinear (resp. sesquilinear for J) form on $E \times F$. Suppose that $E/F^0$ and $F/E^0$ are of finite dimension over A. The rank of $\Phi$ is called the common (finite) dimension of the vector spaces $E/F^0$ and $F/E^0$.

When $E/F^0$ and $F/E^0$ are of infinite dimension, one says that $\Phi$ is of infinite rank.