\begin{definition}[Definition:Sesquilinear Form]
Let $\Bbb F$ be a subfield of $\C$. 
Let $U$ and $V$ be vector spaces over $\Bbb F$. 
A '''sesquilinear form''' is a function $u: U \times V \to \C$ subject to:
:$(1): \quad \forall \alpha \in \Bbb F, x_1, x_2 \in U, y \in V: \map u {\alpha x_1 + x_2, y} = \alpha \map u {x_1, y} + \map u {x_2, y}$
:$(2): \quad \forall \alpha \in \Bbb F, x \in U, y_1, y_2 \in V: \map u {x, \alpha y_1 + y_2} = \bar \alpha \map u {x, y_1} + \map u {x, y_2}$
That is, $u$ is linear in the first argument, and conjugate linear in the second.
If $\Bbb F$ is a subfield of $\R$, then a sesquilinear form is the same as a bilinear map.
\end{definition}