ASSIGNMENT 2: MATH2320/6110
DUE TIME: 5:00PM, 26 MARCH 2024
Question 1. (5 + 5 + 6 = 16 points) Let n ∈ N be a fixed natural number.
(a) Show that the function d : Rn × Rn → R defined by
d(x, y) :=
n∑
i=1
|xi − yi|
1 + |xi − yi|
is a metric on Rn, where x = (x1, · · · , xn), y = (y1, · · · , yn) ∈ Rn.
(b) Can this metric be induced from a norm on Rn? Prove your answer. No marks for
a correct guess without justification.
(c) Show that a subset A ⊂ Rn is open with respect to this metric if and only if it is
open with respect to the Euclidean metric
d2(x, y) =
(
n∑
i=1
|xi − yi|2
)1/2
.
Question 2. (12 points) Let (X, ⟨·, ·⟩) be an inner product space and let ∥ · ∥ be the
induced norm. Let x, y ∈ X be two nonzero elements. Show that the following three
statements are equivalent:
(a) There exist α > 0 such that x = αy.
(b) ⟨x, y⟩ = ∥x∥∥y∥.
(c) ∥x+ y∥ = ∥x∥+ ∥y∥.
Question 3. (6 + 8 + 8 = 22 points) Two norms on a vector space X are said to be
comparable if there exist constants A,B > 0 such that
∥x∥1 ≤ A∥x∥2 and ∥x∥2 ≤ B∥x∥1 for all x ∈ X.
On the vector space
C1[0, 1] := {f ∈ C[0, 1] : f ′ exists and is continuous on [0, 1]} ,
consider the following functions
∥f∥1 := sup
x∈[0,1]
|f(x)|+ sup
x∈[0,1]
|f ′(x)|,
∥f∥2 :=
∫ 1
0
|f(x)|dx+ sup
x∈[0,1]
|f ′(x)|,
∥f∥3 :=
∫ 1
0
|f(x)|dx+
∫ 1
0
|f ′(x)|dx.
(a) Show that ∥ · ∥1, ∥ · ∥2 and ∥ · ∥3 are norms on C1[0, 1].
(b) Are ∥ · ∥1 and ∥ · ∥2 comparable?
(c) Are ∥ · ∥2 and ∥ · ∥3 comparable?
Prove your answer in each case. No marks for correct guesses without justification.