Metric Spaces

Question 1

Define a Metric Space.

Answer 1

Question 2

What is the formula for the Euclidean norm on ℝⁿ or ℂⁿ?

Answer 2

Question 3

What are modulus and the Euclidean norm examples of?

Answer 3

Metrics

Question 4

What is the Euclidean norm on ℝ² called?

Answer 4

The dot product

Question 5

What is the formula for the Euclidean norm on ℂ²?

Answer 5

Question 6

What is the formula for the Euclidean norm on ℝ²?

Answer 6

Question 7

What is the formula for a Metric induced from inner products in vector spcaes?

Answer 7

Question 8

What inequality do you use to check the following metric satisfies the triangle inquality?

Answer 8

Cauchy-Schwarz

Question 9

Define a norm.

Answer 9

Question 10

What is a normed vector space?

Answer 10

A vector space equpped with a norm

Question 11

Norms and inner products in vector spaces are examples of what?

Answer 11

Metrics

Question 12

What is the formula for the l_p-norm?

Answer 12

Question 13

What is a name for the l_p-norm when p=1?

Answer 13

Taxicab norm

Question 14

What is the formula for the l_∞-norm?

Answer 14

Question 15

What is another name for the l_∞-norm?

Answer 15

Sup-norm.

Question 16

What is another name for the Riemannian metric on ℂ?

Answer 16

Chordal metric on ℂ

Question 17

What is the Riemannian metric?

Answer 17

Question 18

What is the Discrete metric

Answer 18

Question 19

What is the subspace metric?

Answer 19

Question 20

What is another name for the sunflower metric?

Answer 20

French railway metric

Question 21

What is the sunflower metric?

Answer 21

Question 22

Prove the following sunflower metric satisfies (D3) the triangle inequality.

Answer 22

Question 23

What is the ‘jungle river’ metric?

Answer 23

Question 24

Prove the following ‘jungle river’ metric is a metric.

Answer 24

Need to do - sheet 7

Question 25

Define open and closed balls in metric spaces.

Answer 25

Question 26

Draw the unit balls for the l₁, l₂, l_∞. What does the l_p-norm look like in comparison?

Answer 26

Question 27

What does collinear mean?

Answer 27

The points lie in a straight line

Question 28

What two cases do you have to consider when drawing balls with respect to the sunflower metric?

Answer 28

1. Collinear
2. Not collinear

Question 29

What do balls with respect to the sunflower metric look like?

Answer 29

Balls when x and y are not collinear and then straight lines where they are.

Question 30

Define open/closed sets in a metric space.

Answer 30

Question 31

What does clopen mean?

Answer 31

When a metric space is open and closed at the same time.

Question 32

Give two common examples of clopen sets.

Answer 32

1. The empty set ∅
2. The whole metric space X

Question 33

What is a lemma about open balls?

Answer 33

Question 34

Prove the following Lemma.

Answer 34

Question 35

Are the following subets of the complex plane open or closed: ℍ, ⅅ, ℂ٭, B_r(z) and ℂ\ℝ_≤0?

Answer 35

Open

Question 36

What is the generic formula for a ball with respect to the discrete metric space?

Answer 36

Question 37

Are balls with respect to the discrete metric open or closed?

Answer 37

Clopen

Question 38

Prove balls with respect to the discrete metric are clopen.

Answer 38

Question 39

Give an example of a set which is neither open or closed?

Answer 39

[0,1)

Question 40

When is a set clopen?

Answer 40

When the set and its complement are both open

Question 41

What is the Lemma about the union/intersection of open sets?

Answer 41

Question 42

Prove the following Lemma.

Answer 42

Question 43

What is the corolllary about closed sets to the following Lemma.

Answer 43

Question 44

Let A be a subset of a metric space (X,d). Define the interior A⁰ of A.

Answer 44

Question 45

Let A be a subset of a metric space (X,d). Define the closure Ā of A.

Answer 45

Question 46

Let A be a subset of a metric space (X,d). Define the boundary ∂A of A.

Answer 46

Question 47

Let A be a subset of a metric space (X,d). Define the exterior A^e of A.

Answer 47

Question 48

Are the interior, closure, boundary and exterior of A closed or open?

Answer 48

1. Interior - Open
2. Closure - Closed
3. Boundary - Closed
4. Exterior - Open

Question 49

Finish the following two statements about the properites of a subset A ⊂ X?

Answer 49

Question 50

Define limits and convergence in a metric space.

Answer 50

Question 51

Finish the following Lemma.

Answer 51

Question 52

Prove part (i) of the following Lemma.

Answer 52

Question 53

Prove part (ii) of the following theorem.

Answer 53

Question 54

What is the key to proving anything to do with limits of sequences in metric spaces?

Answer 54

Write down the definitions in your assumptions and aslo write precisely what you need to prove.

Question 55

Define continuity.

Answer 55

Question 56

What is the Lemma about the basic properties of continuous functions?

Answer 56

Question 57

What is a preimage?

Answer 57

Question 58

What is the theorem about continuity and open/closed sets.

Answer 58

Question 59

Define compact.

Answer 59

Question 60

What is the Lemma about convergence of subsequences?

Answer 60

Question 61

Prove the following Lemma.

Answer 61

Question 62

What is the proposition about clsoed sets and limits of sequences?

Answer 62

Question 63

What is the corollary about closed sets vs. closedness?

Answer 63

Question 64

Prove the following.

Answer 64

Question 65

Prove the following.

Answer 65

Question 66

Prove the following.

Answer 66

Question 67

Define bounded.

Answer 67

Question 68

What is the Lemma about compact sets and boundedness?

Answer 68

Question 69

Prove the following Lemma.

Answer 69

Question 70

What is the Heine-Borel for ℂ theorem?

Answer 70

Question 71

Is the complex plane ℂ compact?

Answer 71

No because the sequence {ik}_k∈ℕ has no convergent subsequence

Question 72

Finish the following Lemma.

Answer 72

Question 73

What is the theorem about the continuous image of a compact set being compact?

Answer 73

Question 74

Prove the following theorem.

Answer 74