4. Inner Product Spaces

Question 1

Define what it means for 〈·, ·〉 to be an inner product on X

Answer 1

Defintion 4.1:
An inner product is a map 〈·, ·〉 : X × X → F such that
* 〈x, y〉 = 〈y, x〉 for all x, y ∈ X;
* 〈x + y, z〉 = 〈x, z〉 + 〈y, z〉 for all x, y, z ∈ X;
* 〈λx, y〉 = λ〈x, y〉 for all x, y ∈ X, λ ∈ F;
* 〈x, x〉 > 0 for all x ∈ X \ {0}

Question 2

Explain how an inner product on X gives rise to a norm
on X.

Answer 2

Proposition 4.9:
An inner product 〈·, ·〉 induces a norm ‖ · ‖ via the formula
‖x‖ = 〈x, x〉1/2 for x ∈ X.

Question 3

Let (X, 〈·, ·〉) be a real or complex inner product space. State what it means for x, y ∈ X to be orthogonal.

Answer 3

Defintion 4.13:
Let X be an inner product space. Two vectors x,y c X are orthogonal if (x,y)=0

Question 4

State and prove Pythagoras’ Theorem.

Answer 4

Propostion 4.14:
Let X be an inner product space.
If x, y ∈ X are orthogonal, then
‖x + y‖2 = ‖x‖2 + ‖y‖2.
Proof. If 〈x, y〉 = 0, then
‖x + y‖2 = 〈x + y, x + y〉
= ‖x‖2 + 〈x, y〉 + conj〈x, y〉 + ‖y‖2
= ‖x‖2 + ‖y‖2

Question 5

State and prove the parallelogram law for an inner product space.

Answer 5

Proposition 4.18:
Let X be an inner product space.
Then, for any x,y c X, |x+y|^2 + |x-y|^2 = 2|x|^2 + 2|y|^2
Proof. Given x,y c X we write |x+-y|^” = (x+-y , x+-y)

Question 6

Carefully state the Cauchy–Schwarz inequality for a general inner product space.

Answer 6

Proposition 4.11: (the Cauchy-Schwarz inequality)
Let X be an inner product space then
|⟨x, y⟩| ≤ ∥x∥∥y∥ for x, y ∈ X.

Question 7

What is the polarisation identity?

Answer 7

Let X be an inner product space over the filed F.
If F=R then ⟨x,y⟩=4 ∥x+y∥ −∥x−y∥ , x,y∈X,
If F=C ⟨x,y⟩=4 ∥x+y∥ −∥x−y∥ +i∥x+iy∥ −i∥x−iy∥ ,x,y∈X.