Let V be a vector space over \(\mathbb{R}\). Inner product is an operation \({\langle \;, \;\rangle : V \times V \rightarrow \mathrm{R} }\) that satisfies the following properties:

$$\forall_{ v,w\in V}: \langle v, w \rangle =\langle w, v\rangle$$

$$\forall_{ v,w \in V; \alpha \in \mathrm{R}}: \; \langle \alpha v , w \rangle = {\alpha} \langle v , w \rangle $$

$$\forall_{ v,w,u\in V }: \; \langle v + w , u \rangle = \langle v , u \rangle + \langle w, u \rangle$$

$$\forall_{ v \in V} \langle v,v\rangle \geq 0 \quad \text{and} \quad \langle v,v \rangle = 0 \iff v=0$$

Let V be a vector space over \(\mathrm{C}\). Hermitian inner product is an operation \({\langle \;, \;\rangle : V \times V \rightarrow \mathrm{C} }\) that satisfies the following properties:

$$\forall_{ v,w\in V}: \langle v, w \rangle = { \langle w, v\rangle}^{*} $$

$$\forall_{ v,w \in V; \alpha \in \mathrm{C}}: \; \langle \alpha v , w \rangle = {\alpha}^{*} \langle v , w \rangle $$

$$\forall_{ v,w,u\in V}: \; \langle v + w , u \rangle = \langle v , u \rangle + \langle w, u \rangle$$

$$\forall_{ v \in V} \langle v,v\rangle \geq 0 \quad \text{and} \quad \langle v,v \rangle = 0 \iff v=0$$