Inner Products (⟨·,·⟩ (·,·) ⟨·|·⟩ (·|·))

an inner product (⟨·,·⟩ : 𝑉×𝑉 → 𝐹) on a vector space (𝑉,𝐹) is a function that takes any pair vectors 𝑢,𝑣∊𝑉 and outputs a real number while satisfying 3 axioms
a vector space (𝑉,𝐹) with an inner product (⟨·,·⟩) is called an inner product space (𝑉,𝐹,⟨·,·⟩)
used to measure how much of one vector is pointing in the direction of another one
an inner product (⟨·,·⟩) induces a norm (||·||⟨·,·⟩) defined as: ||·||_⟨·,·⟩ = √⟨·,·⟩

Inner Product - Definition (3 Axioms)

Where Field (𝐹) is Real Numbers

Where Field (𝐹) is Complex Numbers

For all vectors 𝑢,𝑣,𝑤 in vector space (𝑉,𝐹) and all scalars 𝑐 in field (𝐹)

positive-definiteness	⟨𝑢,𝑢⟩ ≥ 0 ⟨𝑢,𝑢⟩ = 0 if and only if 𝑢 = 0
symmetry	$⟨ u, v ⟩ = ⟨ v, u ⟩$
linearity	⟨𝑢+𝑣,𝑤⟩ = ⟨𝑢,𝑤⟩ + ⟨𝑣,𝑤⟩ ⟨𝑢,𝑣+𝑤⟩ = ⟨𝑢,𝑣⟩ + ⟨𝑢,𝑤⟩ ⟨𝑐𝑢,𝑣⟩ = 𝑐⟨𝑢,𝑣⟩ ⟨𝑢,𝑐𝑣⟩ = 𝑐⟨𝑢,𝑣⟩

positive-definiteness	⟨𝑢,𝑢⟩ ≥ 0 ⟨𝑢,𝑢⟩ = 0 if and only if 𝑢 = 0
conjugate symmetry	$⟨ u, v ⟩ = \overline{⟨ v, u ⟩}$
linearity OF THE FIRST ARGUMENT	⟨𝑢+𝑣,𝑤⟩ = ⟨𝑢,𝑤⟩ + ⟨𝑣,𝑤⟩ ⟨𝑐𝑢,𝑣⟩ = 𝑐⟨𝑢,𝑣⟩

Inner Product - Types

Type	Input	Definition
Dot Product Standard Inner Product	vectors	$⟨ v, u ⟩ = \sum_{i} v_{i} u_{i}$
Frobenius Inner Product	matrices	$⟨ A, B ⟩_{F} = \sum_{i, j} \overline{A_{ij}} B_{ij} = T r (\overline{A^{T}} B) = T r (A^{†} B)$ where the overline denotes the complex conjugate, and † denotes the Hermitian conjugate
Inner Product of Functions	functions	$⟨ f (x), g (x) ⟩_{a, b} = \int_{a}^{b} f (x) g (x) d x on the interval of [a, b]$
L2 Inner Product	functions	$⟨ f, g ⟩_{L^{2}} = \int_{X} f g d 𝜇$

Inner Product - Properties