1) Metric Spaces

Question 1

Metric space defn

Answer 1

Definition 1.1 (Metric Space). A metric space (X, d) is a set X together with a function d : X × X → R that satisfies the following properties
(i) d(x, y) ⩾ 0; and d(x, y) = 0 ⇐⇒ x = y (positive definite); (ii) d(x, y) = d(y, x) (symmetric);
(iii) d(x,z)⩽d(x,y)+d(y,z)(triangleinequality).

The function d is called the metric.

Question 2

Standard distance between two reals

Answer 2

d(x, y) = |x − y|.

Question 3

Distance between two points on a plane

Answer 3

d(x, y) = d((x_1, x_2), (y_1, y_2)) = √[(x_1 − y_1)^2 + (x_2 − y_2)^2].

x,y vectors

Question 4

E.g 1.2 standard metric

Answer 4

X = R. The standard metric is given by d1(x, y) = |x − y|.

Question 5

E.g of metrics
Exponential one -proof

Generic of discrete -proof

Answer 5

d(x,y)=|e^x −e^y|;

d(x, y) =
|x−y| if|x−y|⩽1,
1 if |x − y| ⩾ 1.

Question 6

Discrete metric

Answer 6

d(x,y) =
{1 ifx̸=y,
{0 ifx=y.

Question 7

The Euclidean metric in R^n

Answer 7

If X= R^m. The standard metric

x
= (x_1, …,x_m)

y
= (y_1, …,y_m)
Then
d_2(x,y)=sqrt[(x_1 −y_1)^2 +(x_2 −y_2)^2 +…+(x_m −y_m)^2]

Question 8

Euclidean metric links

Answer 8

linked to the inner-product (scalar product),
x · y = x_1y_1 + x_2y_2 + . . . + x_my_m, since it is just sqrt[(x − y).(x − y)]

Question 9

d_∞(x,y) in R^m

Answer 9

d∞(x,y)=
max{|x1 −y1|,|x2 −y2|,…,|xm −ym|}.

Question 10

metric on R^m
d_1

Answer 10

d_1(x,y)=|x1 −y1|+|x2 −y2|+…+|xm −ym|.

Question 11

d1 , d2 , d∞ are all ____ and ____

Answer 11

d1 , d2 , d∞ are all translation-invariant

d(x + z, y + z) = d(x, y)),

and positively homogeneous (i.e., d(kx, ky) = |k|d(x, y)

Question 12

Metrics
For X = C[a,b]

d1 , d2 , d∞

Answer 12

d_2 (f, g)
= √ [∫_[a,b] |f(x) − g(x)|^2 dx]].
a
Again, this is linked to the idea of an inner product,

d_1(f, g) =

[∫_[a,b] |f(x) − g(x)|dx]].

the area between two graphs

d∞(f, g) = max{|f(x) − g(x)| : a ⩽ x ⩽ b}, the maximum vertical separation between two graphs.

Question 13

Example 1.3. On C[0, 1] take
f(x) = x and g(x) = x^2 and calculate
d_1 (f,g)

Answer 13

d1(f, g) = the area between two graphs ∫_[b, a]
|f(x) − g(x)| dx
= [0.5x^2 - (1/3)x^3] on interval [0,1]
= 1/6

Question 14

Example 1.3. On C[0, 1] take
f(x) = x and g(x) = x^2 and calculate
d_2 (f,g)

Answer 14

d_2 (f, g)
= √ [∫_[a,b] |f(x) − g(x)|^2 dx]].

(x-x^2)^2 = x^2 +x^4 -2x^3
= (1/3)x^3 + 0.2x^5 -0.5x^4

= √(1/30)

Question 15

Example 1.3. On C[0, 1] take
f(x) = x and g(x) = x^2 and calculate
d_ ∞(f,g)

Answer 15

d∞(f, g) = max{|f(x) − g(x)| : a ⩽ x ⩽ b}, the maximum vertical separation between two graphs.

d∞(f,g) =
max _{x∈ [0,1]} |x−x^2|= 1/4
(* y=x-x^2 , dy /dx= 1-2x has turning point x=0.5, y= 0.25 maximum second deriv =-2 *
1/4.

Question 16

Remark 1.4 subsets of metric spaces

Answer 16

Any subset of a metric space is again a metric space its own right, by restricting the distance function to the subset.

Question 17

Example 1.5. a subspace of R.

Answer 17

(i) The interval [a, b] with d(x, y) = |x − y| is a subspace of R.

Question 18

Unit circle on R2.
With d2

Answer 18

The unit circle
{(x1,x2) ∈ R^2 : (x_1)^2 + (x_2)^2 = 1} with
d_2(x,y) = sqrt[(x1 − y1)^2 + (x2 − y2)^2 ]is a SUBSPACE of R2.

d_2(0,x) ≤ 1 sqrt( x_1 ^2 + x_2 ^2) ≤ 1

Looks like a circle

Question 19

Metric space P

Answer 19

The space of polynomials P is a metric space with any of the metrics inherited from C[a, b] above

Question 20

Useful applications of MS properties

Answer 20

d(x,y) = d(x-y,y-y) = d(x-y, o)

d(x,y) - d(a,b)| <= d(x,a) + d(y,b)

Question 21

Unit circle d_1 in R^2

Answer 21

SUBSPACE of R2.

d_1(0,x) ≤ ( x_1 + x_2 ) ≤ 1

Looks like a rhombus
Contained in unit circles in d_ inf and s
D_2

Question 22

Unit balls

Answer 22

{x ∈V: d(0,x) q } = B₁ (x)

Question 23

Unit circle on d∞

Answer 23

d∞(0,x)<= 1

max{x₁,x₂} <= 1

d((x_1, x_2), (y_1, y_2)) = max {|x_1 -y_1|, |x_2 - y_2| } what does the “unit circle” and “unit disk” about (0,0) look like?

Now one shouldn’t get the wrong impression. SOMETIMES one can get a visual image of what a disk (or “ball” in higher dimensions) and a circle (or “sphere”) look like. But sometimes visualizing is impossible.

Square

Question 24

Unit balls relationship

Answer 24

Ball d₁ ⊆ Ball d₂ ⊆ Ball d∞

Question 25

If has the property of closed under + and scalar multiplication

Answer 25

Vector space
Can be represented by basis #dimensions