Description

This assignment has a total of (18Marks).
Exercise 2.1

i) Let (V,⟨·,·⟩real polarisation identityR) be a real inner product space and: ∥x∥ = √⟨x,x⟩ the norm induced by the inner product. Prove the

(2Marks)

ii) Let (V,⟨·,·⟩complex polarisation identityC) be a complex inner product space and: ∥x∥ = √⟨x,x⟩ the norm induced by the inner product. Prove the

(2Marks) iii) Let V be a real or complex vector space. Show that every norm on V , if it is induced by some inner product, satsfies the parallelogram rule:
∥x + y∥2 + ∥x − y∥2 = 2(∥x∥2 + ∥y∥2) for all x,y ∈ V
(2Marks)
iv) Prove that the norm is not induced by an inner product, i.e., there

exists no inner product ⟨·,·⟩ such that ∥·∥∞ = √⟨·,·⟩.
(2Marks)
v) Show that every norm that satisfies the parallelogram rule is induced by an inner product. For simplicity, consider a real vector space only. Instructions:
• Use the polarization identity to define an inner product from the norm.
• Show that the so-defined inner product satisfies ⟨x,y + z⟩ = ⟨x,y⟩ + ⟨x,z⟩.
• Then deduce that ⟨x,λy⟩ = λ⟨x,y⟩ for rational λ ∈ Q.
• Use the continuity of the norm to conclude that the equality holds in fact for λ ∈ R.
• Verify the other properties for an inner product.
(4Marks)
Exercise 2.2
We define the following spaces of complex-valued sequences (an)n∈N:
, c0 = {(an)n∈N: nlim→∞an = 0},
and norms
, .
i) (3Is ℓMarks1 dense in) c0 in the ∥·∥∞ norm? Why or why not? Explain! ii) Is(3ℓMarks1 dense in) c0 in the ∥·∥1 norm? Why or why not? Explain!