Sesquilinearity

linear-algebra

Definition

Sesquilinearity

A map $B:V\times V\to F$ on a vector space $V$ over a field $F$ is sesquilinear if it is linear in one argument and semilinear in the other. The two assignments are equivalent via conjugate symmetry.

Mathematical Convention $x,y,z\in V$ and $\alpha ,\beta \in F$:

The mathematical convention makes the first argument linear, i.e. for all

$$\begin{array}{rl}B(\alpha x+\beta y,z)& =\alpha B(x,z)+\beta B(y,z),\\ B(x,\alpha y+\beta z)& =\overline{\alpha }B(x,y)+\overline{\beta }B(x,z).\end{array}$$

Bra-Ket (Physics) Convention bra-ket notation (and in physics) the opposite convention is used: the ket (second argument) is the linear side, the bra is the semilinear side:

In

$$\begin{array}{rl}B(\alpha x+\beta y,z)& =\overline{\alpha }B(x,z)+\overline{\beta }B(y,z),\\ B(x,\alpha y+\beta z)& =\alpha B(x,y)+\beta B(x,z).\end{array}$$