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 the 1st 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:
$$\forall x\in\mathbb{R},\; x=\overline{x}$$
Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.
Note: confusingly, the convention in physics is often the exact opposite: an inner product is linear in the second slot and anti-linear in the first slot.
Derived properties
The inner product is anti-linear in the second slot
$$\begin{gathered} \forall u,v,w\in V, a \in \mathbb{F}: \\ \begin{cases} \langle u, v+w \rangle = \langle u, v \rangle+\langle u, w \rangle \\ \langle u, av \rangle = \overline{a}\langle u, v \rangle \end{cases} \end{gathered}$$
Proof:
For additivity:
$$\begin{gathered} \langle u, v+w \rangle = \overline{\langle v+w, u \rangle} = \overline{\langle v, u \rangle+\langle w, u \rangle} \\ = \overline{\langle v, u \rangle} + \overline{\langle w, u \rangle} = \langle u, v \rangle + \langle u, w \rangle \end{gathered}$$
Similarly, for anti-homogeneity:
$$\langle u, av \rangle = \overline{\langle av, u \rangle} = \overline{a\langle v, u \rangle} = \overline{a} \overline{\langle v, u \rangle} = \overline{a} \langle u, v \rangle$$