1) Definitions and Examples

Question 1

What is a metric space

Answer 1

Question 2

What is d1, d2 and d∞

Answer 2

Question 3

Describe the proof that d2 is a metric in Euclidean n-space

Answer 3

Question 4

Describe the proof that d1 is a metric in Euclidean n-space

Answer 4

Question 5

Describe the proof that d∞ is a metric in Euclidean n-space

Answer 5

Question 6

Prove that for any metric space (X,d) that Br(x) ⊆ B_r(x)

Answer 6

Question 7

What is an open ball and a closed ball

Answer 7

Question 8

What are the open and closed balls in the d2 metric

Answer 8

Question 9

Describe the pictures of the set inclusions of the three main metrics

Answer 9

• The balls for d1 are diamonds
• The balls for d2 are discs
• The balls for d∞ r(x) are squares

Question 10

What is the criteira for two metrics to be Lipschitz Equivalent

Answer 10

Question 11

In terms of inequalities, how are d1, d2 and d∞ related

Answer 11

Question 12

Describe the proof that

Answer 12

Question 13

What is the discrete metric

Answer 13

Question 14

Describe the proof that the discrete metric is a metric

Answer 14

• Axiom 1 qnd 2 are satisfied automatically.
• Axiom 3 - For any x,y,z in X, d(x,y)+d(y,z) is either 0, 1, or 2. Since d(x,z) is either 0 or 1, d(x,z)≤d(x,y)+d(y,z) always holds

Question 15

What is an isomerty between metric spaces

Answer 15

Question 16

What is the standard metric dc on the complex numbers

Answer 16

Question 17

What is a subspace

Answer 17

A subset of a metric space that itself forms a metric space under the same metric

Question 18

What does it mean for a function to be bounded, continous and polynomial of order n

Answer 18

• Bounded - There exists a constant K such that |f(x)| ≤ K for every x ∈ [a, b],
• Continuous - ∀c ∈ [a, b], ∀ε > 0, ∃ δ > 0 : ∀x ∈ [a, b], |x−c| < δ =⇒ |f(x)−f(c)| < ε
• Polynomial of degree n - It can be written as a(x) = anx^n + · · · + a1x + a0 ,

Question 19

What is the inclusion relationship between Poly[a,b], Cont[a,b] and Bdd[a,b]

Answer 19

Poly[a,b] ⊆ Cont[a,b] ⊆ Bdd [a,b]

Question 20

What is dsup(f,g)

Answer 20

A metric on the Bdd[a,b], Cont[a,b] and Poly[a,b]

Question 21

What does an open ball f(x) = x^2 on [−1, 1] with radius 1 look like

Answer 21

Question 22

What is the L1[a,b] metric space

Answer 22

The metric d1 on Cont[a,b]

Question 23

What is the L2[a,b] metric space

Answer 23

The metric d2 on Cont[a,b]

Question 24

What is the Cartesian Product of (X,d) and (Y,e)

Answer 24