Product space

$Norm$

∥ \cdot ∥

is a norm:

\forall x, y \in V, \forall α \in K

(N 1) ∥ x ∥ \geq 0; ∥ x ∥ = 0 \Leftrightarrow x = 0 (N 2) ∥ αx ∥ = ∣ α ∣∥ x ∥ (N 3) ∥ x + y ∥ \leq ∥ x ∥ + ∥ y ∥

(V, ∥ \cdot ∥)

is a notm space:

∥ \cdot ∥

is a norm

⟨ \cdot \cdot ⟩

is a inner product:

\forall x, y, z \in V, α \in K

1. ⟨ x, x ⟩ \geq 0, ⟨ x, x ⟩ = 0 \Leftrightarrow x = 0 2. ⟨ x, y ⟩ = \overline{⟨ y, x ⟩} 3. ⟨ αx, y ⟩ = α ⟨ x, y ⟩ 4. ⟨ x + y, z ⟩ = ⟨ x, z ⟩ + ⟨ y, z ⟩

(V, ⟨ \cdot \cdot ⟩)

is an Inner product space :

⟨ \cdot \cdot ⟩

is a inner product

1. ⟨ x, α y ⟩ = \overline{α} ⟨ x, y ⟩ 2. ⟨ x, y + z ⟩ = ⟨ x, y ⟩ + ⟨ x, z ⟩

Norm of inner product space

V

:

∥ x ∥ = ⟨ x, x ⟩

∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥∥ y ∥