# Definition A sesquilinear form $\braket{.|.}$ is a [[Quadratic Forms on Vector Spaces|quadratic form]] on a [[vector space]] $V$ $ \begin{align} \braket{.|.}: &V\times V \rightarrow \mathbb{F} , \\ &(\pmb{x}, \pmb{y}) \mapsto \braket{\pmb{x}|\pmb{y}} \end{align} $ where $\mathbb{F}$ is the field over which $V$ is defined. It has the following properties (in physicists' convention) 1. *Linearity in its second argument:* $\braket{\pmb{z}|c\pmb{x}+\pmb{y}} = c\braket{\pmb{z}|\pmb{x}} + \braket{\pmb{z}|\pmb{y}}$ 2. *Antilinearity in its first argument:* $\braket{c\pmb{x}+\pmb{y}|\pmb{z}} = c^* \braket{\pmb{x}|\pmb{z}} + \braket{\pmb{y}|\pmb{z}}$