Definition

$⟨ \cdot, \cdot ⟩ : V \times V \to K$ A function with the following properties

$\forall u, v, w \in V : ⟨ u + v, w ⟩ = ⟨ u, w ⟩ + ⟨ v, w ⟩$ ^[Additivity]
$\forall u, v \in V, k \in K : ⟨ k u, v ⟩ = k ⟨ u, v ⟩$ ^[Homogeneity]
$\forall u, v \in V : ⟨ u, v ⟩ = \overline{⟨ v, u ⟩}$ ^[Conjugate symmetry]
$\forall 0 \neq = v \in V : ⟨ v, v ⟩ > 0$ ^[Positive definiteness] where $K \in {R, C}$ on a vector space $(V, +, \cdot, K)$

Vector

for $x, y \in R^{n}$ $⟨ x, y ⟩ = ∣∣ x ∣ \cdot ∣∣ y ∣∣ cos θ = \sum_{i = 1}^{n} x_{i} y_{i}$ where $∣∣ \cdot ∣∣$ is a Norm

for $x, y \in C^{n}$ $⟨ x, y ⟩ = x \overline{y} = \sum_{i = 1}^{n} x_{i} \overset{y}{ˉ}_{i}$

$⟨ f, g ⟩ = \int_{a}^{b} f (x) g (x) d x$ where $f, g \in C_{p} (a, b)$

$⟨ f, g ⟩ = ⟨ g, f ⟩$ $⟨ f, g + h ⟩ = ⟨ f, g ⟩ + ⟨ f, h ⟩$ $⟨ c f, g ⟩ = c ⟨ f, g ⟩, c \in R$ $⟨ f, f ⟩ \geq 0$ (by the definition of Norm)

$x^{⊺} y = 0 \Rightarrow x ⊥ y$ $x^{⊺} y < 0 \Rightarrow π > θ > \frac{1}{2} π$ $x^{⊺} y > 0 \Rightarrow 0 < θ < \frac{1}{2} π$

$\overline{x} y \in / R \Rightarrow \overline{\overline{x} y} \neq = \overline{y} x$

The inner product of Euclidean Space is called a dot product.