Inner product
An inner product on $V$ is a map
$$ \langle \cdot, \cdot \rangle : V \times V \to \mathbb{F}, \quad (u, v) \mapsto \langle u, v \rangle $$
It must satisfies four properties:
- Linearity in first slot: $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ and $\langle au, v \rangle = a \langle u, v \rangle$ for all $u, v, w \in V; a \in \mathbb{F}$.
- Positivity: $\langle v, v \rangle \geq 0$ for all $v \in V$.
- Positive definiteness: $\langle v, v \rangle = 0$ if and only if $v = 0$.
- Conjugate symmetry: $\langle u, v \rangle = \overline{\langle v, u \rangle}$ for all $u, v \in V$.
Recall that every real number $x \in \mathbb{R}$ equals its complex conjugate. Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.