Norms 1 - 4

Question 1

Define a norm on a vector space X

Answer 1

Question 2

Define the standard euclidean norm in Rⁿ

Answer 2

Question 3

Define a normed space

Answer 3

A pair (X, || . ||) where || . || is a norm and X is a vector space

Question 4

Define convex on a vector space

Answer 4

Question 5

Define the closed unit ball B_X and prove that it’s convex in a normed space

Answer 5

Question 6

Answer 6

Question 7

Define convex on a function f:[a,b] to R

Answer 7

Question 8

Define the l^p norms

Answer 8

Question 9

State and prove Minkowski’s inequality in Rⁿ

Answer 9

Question 10

When are two norms equivalent

Answer 10

Question 11

Define the l^p sequence space

Answer 11

Question 12

What do we denote by C[(a,b)]?

Answer 12

the space of real-valued continuous functions on the interval [a,b]

Question 13

What norm do we use normally on C( [a,b] )

Answer 13

Question 14

Define a metric d on a set X

Answer 14

Question 15

Answer 15

Question 16

Define the discrete metric on X

Answer 16

d(x,x) = 0 and d(x,y) = 1 if x isnt equal to y

Question 17

Define the open ball B(a,r) and the closed ball

Answer 17

Question 18

When is a subset S of (X,d) bounded

Answer 18

if there exist a in X and r > 0 such that S is a subset of B(a, r)

Question 19

Show that if A is a bounded subset of (X,d) there is an a in A and r > 0 such that A is a subset of B(a, r)

Answer 19

Question 20

Define open and closed for a subset U of (X,d)

Answer 20

Question 21

Lemma: Open balls are open

Answer 21

Question 22

Lemma: If U₁, ……., U_n are open then the intersection is open

Answer 22

Question 23

Lemma: If U₁,……, U_n are open then the union is open

Answer 23

Question 24

When does a sequence (x_n) converge to some x in X

Answer 24

Question 25

Lemma: A sequence can have at most one limit

Answer 25

Question 26

Answer 26

Question 27

Lemma: A subset of a metric space is open if and only if it’s the union of open balls

Answer 27

Question 28

Define continuity for a function f: X to Y on metric spaces

Answer 28

Question 29

Answer 29

Question 30

Answer 30

Question 31

Lemma: Suppose that X,Y,Z are metric spaces and f: X to Y, g: Y to Z are continuous. Then gf: X to Z is continuous

Answer 31

If U is an open subset of Z then g^-1(U) is open in Y. So f^-1(g^-1(U)) is open in X

Question 32

Answer 32

The functions f(x,y) = y and g(x,y) = x - 7y are continuous, so U = f^-1(0, inf) intersect g^-1(0, inf) is open

Question 33

When are two metrics topologically equivalent

Answer 33

Two metrics d₁ and d₂ are called topologically equivalent if the open sets (X, d₁) and (X, d₂) coincide

Question 34

When is f: X to Y an isometry between X and Y

Answer 34

When d_y(f(x), f(y)) = d_x(x,y)

Question 35

When is f: X to Y a homeomorphism?

Answer 35

If f is a bijection such that f and f^-1 are continuous. U is open in X if and only if f(U) is open in Y

Question 36

Define a topological property of a metric space (X,d)

Answer 36

A topological property is a property such that every metric space homeomorphic to (X,d) has the same property

Question 37

Give 4 examples of topological properties

Answer 37

• X is open, X is closed
• X is finite, countable, uncountable
• every subset of X is open
• every continuous real valued function on X is bounded