Inner Product Space

Definition: Complex Inner Product Space

A complex inner product space is a complex vector space $(V, F, +, \cdot)$ equipped with an inner product operation $⟨ \cdot, \cdot ⟩ : V \times V \to C$ which has the following properties:

Conjugate symmetry

$⟨ u, v ⟩ = \overline{⟨ v, u ⟩} \forall u, v \in V$

Sesquilinearity

Distributivity I: $⟨ u + v, w ⟩ = ⟨ u, w ⟩ + ⟨ v, w ⟩ \forall u, v, w \in V$

Distributivity II: $⟨ u, v + w ⟩ = ⟨ u, v ⟩ + ⟨ u, w ⟩ \forall u, v, w \in V$

Semi-linearity in the first argument: $⟨ λ u, v ⟩ = \overset{ˉ}{λ} ⟨ u, v ⟩ \forall u, v \in V, \forall λ \in C$

Linearity in the second argument: $⟨ u, λ v ⟩ = λ ⟨ u, v ⟩ \forall u, v \in V, \forall λ \in C$

Positive-definiteness

$⟨ v, v ⟩ \geq 0 with ⟨ v, v ⟩ = 0 ⟺ v = 0 \forall v \in V$

Definition: Real Inner Product Space

A real inner product space is a real vector space $(V, F, +, \cdot)$ equipped with an inner product operation $⟨ \cdot, \cdot ⟩ : V \times V \to R$ which has the following properties:

Symmetry

$⟨ u, v ⟩ = ⟨ v, u ⟩ \forall u, v \in V$

Sesquilinearity

Distributivity I: $⟨ u + v, w ⟩ = ⟨ u, w ⟩ + ⟨ v, w ⟩ \forall u, v, w \in V$

Distributivity II: $⟨ u, v + w ⟩ = ⟨ u, v ⟩ + ⟨ u, w ⟩ \forall u, v, w \in V$

Linearity in the first argument: $⟨ λ u, v ⟩ = λ ⟨ u, v ⟩ \forall u, v \in V, \forall λ \in C$

Linearity in the second argument: $⟨ u, λ v ⟩ = λ ⟨ u, v ⟩ \forall u, v \in V, \forall λ \in C$

Positive-definiteness

$⟨ v, v ⟩ \geq 0 with ⟨ v, v ⟩ = 0 ⟺ v = 0 \forall v \in V$

NOTATION

We write $(V, F, +, \cdot)$ when it does not matter if the inner product space is real or complex.

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Inner Product Space

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