# Definition:Sesquilinear Form

## Definition

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.

## Linguistic Note

Sesqui means one-and-a-half, deriving from the Latin for and also a half.

This gives rise to the term sesquilinear if one regards conjugate linearity as being almost or half linearity.