Definition 12.6.1 (Sesquilinear Form).label Let $E$ be a vector space over $K \in \RC$ and $\lambda: E \times E \to \complex$, then $\lambda$ is a sesquilinear form if:

1. (1)

   For each $x, y, z \in E$ and $\mu \in K$, $\lambda(\mu x + y, z) = \mu\lambda(x, z) + \lambda(y, z)$.

2. (2)

   For each $x, y, z \in E$ and $\mu \in K$, $\lambda(x, \mu y + z) = \ol \mu\lambda(x, y) + \lambda(x, z)$.