Definition:Sesquilinear Form

Definition
Let $\HH, \KK$ be Hilbert spaces over $\Bbb F \in \set {\R, \C}$.

A sesquilinear form is a function $u: \HH \times \KK \to \Bbb F$ subject to:


 * $(1): \quad \forall \alpha \in \Bbb F, h_1, h_2 \in \HH, k \in \KK: \map u {\alpha h_1 + h_2, k} = \alpha \map u {h_1, k} + \map u {h_2, k}$
 * $(2): \quad \forall \alpha \in \Bbb F, h \in \HH, k_1, k_2 \in \KK: \map u {h, \alpha k_1 + k_2} = \bar \alpha \map u {h, k_1} + \map u {h, k_2}$

That is, $u$ is linear in the first argument, and conjugate linear in the second.

If $\Bbb F = \R$, then a sesquilinear form is the same as a bilinear map.

Linguistic Note
Sesqui means one-and-a-half.

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