# Definition Given a set $X$, a metric $d: X \times X \rightarrow \mathbb{R}$ is a function that satisfies the conditions, $\forall x,y,z \in X$ 1. *Positive semi-definite:* $d(x,y) \geq 0$ 2. *Identity of Indiscernibles:* $d(x,y) = 0 \iff x = y$ 3. *Symmetry:* $d(x,y) = d(y,x)$ 4. *Triangle Inequality:* $d(x,y) \leq d(x,z) + d(z,y)$ ## Linear Algebra A Hermitian 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:* $\braket{c\pmb{x}+\pmb{y}|\pmb{z}} = c^* \braket{\pmb{x}|\pmb{z}} + \braket{\pmb{y}|\pmb{z}}$ 3. *Hermitian:* $\braket{\pmb{x}|\pmb{y}} = (\braket{\pmb{y}|\pmb{x}})^*$