Sesquilinearity

linear-algebra

Definition

Sesquilinearity

A map $B:V\times V\to F$ on a vector space $V$ over a field $F$ is called sesquilinear if it is linear in one argument and semilinear in the other.

In the complex case, this usually means:

$$B(\alpha x+\beta y,z)=\alpha B(x,z)+\beta B(y,z)$$

and

$$B(x,\alpha y+\beta z)=\overline{\alpha }B(x,y)+\overline{\beta }B(x,z).$$

For example, the inner product on a complex inner product space is sesquilinear.