Triangle inequality

For any vectors $x, y$ in an inner product space:

∥ x + y ∥ \leq ∥ x ∥ + ∥ y ∥ .

Derivation from Cauchy-Schwarz inequality
We compute:

∥ x + y ∥^{2} = ⟨ x + y, x + y ⟩ = ∥ x ∥^{2} + 2 ℜ ⟨ x, y ⟩ + ∥ y ∥^{2} .

Apply Cauchy–Schwarz:

ℜ ⟨ x, y ⟩ \leq | ⟨ x, y ⟩ | \leq ∥ x ∥ ∥ y ∥ .

So,

∥ x + y ∥^{2} \leq ∥ x ∥^{2} + 2 ∥ x ∥ ∥ y ∥ + ∥ y ∥^{2} = (∥ x ∥ + ∥ y ∥)^{2} .

Taking square roots gives:

∥ x + y ∥ \leq ∥ x ∥ + ∥ y ∥ .