Inner product
Mathematical definition
Let $V$ be a vector space over $\mathbb{F}$.
Map
$$ \begin{aligned} \langle \cdot, \cdot \rangle : V \times V \to \mathbb{F}\\ \quad (u, v) \mapsto \langle u, v \rangle \end{aligned} $$
is an inner product on $V$ if 4 conditions are satisfied:
- Linearity in the 1st slot: $\forall (u, v, w \in V, a \in \mathbb{F}), \langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ and $\langle au, v \rangle = a \langle u, v \rangle$
- Definiteness: $\langle v, v \rangle = 0 \iff v = \vec 0$.
- Positivity: $\forall v \in V, v \neq \vec 0, \langle v, v \rangle > 0$
- Conjugate symmetry: $\forall u, v \in V, \langle u, v \rangle = \overline{\langle v, u \rangle}$.
Recall:
$$\forall x\in\mathbb{R},\; x=\overline{x}$$
Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.
Warning: 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:
$$\langle u, av \rangle = \overline{\langle av, u \rangle} = \overline{a\langle v, u \rangle} = \overline{a} \overline{\langle v, u \rangle} = \overline{a} \langle u, v \rangle$$