Exercise 6.1.19

Answers

1.

Just compute

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

= ∥ x ∥^{2} + ⟨ x, y ⟩ + \bar{⟨ y, x ⟩} + ∥ y ∥^{2} = ∥ x ∥^{2} + ℝe (⟨ x, y ⟩) + ∥ y ∥^{2}

and

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

= ∥ x ∥^{2} - ⟨ x, y ⟩ - \bar{⟨ y, x ⟩} + ∥ y ∥^{2} = ∥ x ∥^{2} - ℝe (⟨ x, y ⟩) + ∥ y ∥^{2} .

2.

We have

∥ x - y ∥ + ∥ y ∥ \geq ∥ x ∥

and

∥ x - y ∥ + ∥ x ∥ = ∥ y - x ∥ + ∥ x ∥ \geq ∥ y ∥ .

Combining these two we get the desired inequality.

jephianlin

2011-06-27 00:00