Definition:Sesquilinear Form

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Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$.

A sesquilinear form is a function $u: H \times K \to \Bbb F$ subject to:

$(1): \qquad \forall \alpha \in \Bbb F, h_1, h_2 \in H, k \in K: u \left({\alpha h_1 + h_2, k}\right) = \alpha u \left({h_1, k}\right) + u \left({h_2, k}\right)$
$(2): \qquad \forall \alpha \in \Bbb F, h \in H, k_1, k_2 \in K: u \left({h, \alpha k_1 + k_2}\right) = \bar{\alpha} u \left({h, k_1}\right) + u \left({h, k_2}\right)$

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.

Origin of Name

Sesqui means one-and-a-half. This gives rise to the term sesquilinear if one regards conjugate linearity as being almost or half linearity.