STUDY NOTES & REFERENCE GUIDE
courseroomsity - Stuvia | Page 1 of 691
, STUDY NOTES & REFERENCE GUIDE
Contents
Preface vii
A Note on the Convention xi
About the Author xiii
1. Metric Spaces 1
1.1 Definitions and Examples.................................................................. 1
Exercise 1.1: Part A.....................................10
Exercise 1.1: Part B.....................................14
1.2 Topology of Metric Spaces ............................................................. 36
Exercise 1.2: Part A.....................................50
Exercise 1.2: Part B.....................................64
1.3 Compactness ..................................................................................... 85
Exercise 1.3: Part A.....................................90
Exercise 1.3: Part B.....................................94
1.4 Compactness in the Euclidean Space Rn .................................................. 108
Exercise 1.4: Part A.................................. 115
Exercise 1.4: Part B.................................. 118
2. Limits and Continuity 129
2.1 Convergence in a Metric Space .................................................... 129
Exercise 2.1: Part A.................................. 134
Exercise 2.1: Part B.................................. 138
2.2 Complete Metric Spaces ............................................................... 145
Exercise 2.2: Part A.................................. 150
Exercise 2.2: Part B.................................. 155
2.3 Continuity and Homeomorphism ................................................. 172
Exercise 2.3: Part A.................................. 193
Exercise 2.3: Part B.................................. 204
3. Connectedness 233
3.1 Connectedness ................................................................................ 233
Exercise 3.1: Part A.................................. 245
Exercise 3.1: Part B.................................. 249
xv
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, STUDY NOTES & REFERENCE GUIDE
xvi Metric Space Topology: Examples, Exercises and
Solutions
3.2 Path-connectedness ........................................................................266
Exercise 3.2: Part A....................................278
Exercise 3.2: Part B....................................281
4. Uniform Continuity 295
4.1 Uniform Continuity .......................................................................296
Exercise 4.1: Part A....................................301
Exercise 4.1: Part B....................................309
4.2 Contraction and Banach’s Fixed Point Theorem .....................322
Exercise 4.2: Part A....................................330
Exercise 4.2: Part B....................................332
5. Uniform Convergence 349
5.1 Sequence of Functions ...................................................................349
Exercise 5.1: Part A....................................368
Exercise 5.1: Part B....................................377
5.2 Series of Functions ........................................................................389
Exercise 5.2: Part A....................................395
Exercise 5.2: Part B....................................401
Bibliography 421
Index 423
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, STUDY NOTES & REFERENCE GUIDE
Chapter 1
Metric Spaces
·
In this chapter, the basic concept of metric spaces will be introduced.
Naively, they are simply nonempty sets equipped with a structure
called metric. For the less matured students, at the beginning, this
structure may appear to be a bit abstract and difficult to master.
But in practice, this seemingly new concept is nothing more than
a tiny little abstractization of the familiar space Rn and so all one
needs to do is that whenever one needs to work on a problem in an
abstract metric space, one first looks at the problem on Rn, then one
would be able to see the clue of how to proceed in the general case.
In fact, in general, the most effective way to master a new concept
in any branch of mathematics is to keep in mind a couple of typical
concrete examples and think of these examples all the time. It is just
that easy.
1.1 Definitions and Examples
Definition 1.1.1. Let X be a nonempty set.
A metric on X is a real-valued function
d : X ×X → R
satisf
ying
(M1) d(x, y) ≥ 0 and d(x, y ) = 0 if and only
if x = y,
(M2) (symmetry) d(x, y ) = d(y, x),
(M3) (triangle inequality) d(x, y) ≤ d(x, z)
+ d(z, y)
for all x, y, z ∈ X. Given x, y ∈ X, d(x, y)
is also known as the distance between x and
y with respect to d. The pair (X, d) is
called a metric space and elements in X are
referred to as points in X. For the sake of
convenience, in case there is a clearly
defined metric d on X, we shall simply call
X a metric space.
courseroomsity - Stuvia | Page 4 of 691
courseroomsity - Stuvia | Page 1 of 691
, STUDY NOTES & REFERENCE GUIDE
Contents
Preface vii
A Note on the Convention xi
About the Author xiii
1. Metric Spaces 1
1.1 Definitions and Examples.................................................................. 1
Exercise 1.1: Part A.....................................10
Exercise 1.1: Part B.....................................14
1.2 Topology of Metric Spaces ............................................................. 36
Exercise 1.2: Part A.....................................50
Exercise 1.2: Part B.....................................64
1.3 Compactness ..................................................................................... 85
Exercise 1.3: Part A.....................................90
Exercise 1.3: Part B.....................................94
1.4 Compactness in the Euclidean Space Rn .................................................. 108
Exercise 1.4: Part A.................................. 115
Exercise 1.4: Part B.................................. 118
2. Limits and Continuity 129
2.1 Convergence in a Metric Space .................................................... 129
Exercise 2.1: Part A.................................. 134
Exercise 2.1: Part B.................................. 138
2.2 Complete Metric Spaces ............................................................... 145
Exercise 2.2: Part A.................................. 150
Exercise 2.2: Part B.................................. 155
2.3 Continuity and Homeomorphism ................................................. 172
Exercise 2.3: Part A.................................. 193
Exercise 2.3: Part B.................................. 204
3. Connectedness 233
3.1 Connectedness ................................................................................ 233
Exercise 3.1: Part A.................................. 245
Exercise 3.1: Part B.................................. 249
xv
courseroomsity - Stuvia | Page 2 of 691
, STUDY NOTES & REFERENCE GUIDE
xvi Metric Space Topology: Examples, Exercises and
Solutions
3.2 Path-connectedness ........................................................................266
Exercise 3.2: Part A....................................278
Exercise 3.2: Part B....................................281
4. Uniform Continuity 295
4.1 Uniform Continuity .......................................................................296
Exercise 4.1: Part A....................................301
Exercise 4.1: Part B....................................309
4.2 Contraction and Banach’s Fixed Point Theorem .....................322
Exercise 4.2: Part A....................................330
Exercise 4.2: Part B....................................332
5. Uniform Convergence 349
5.1 Sequence of Functions ...................................................................349
Exercise 5.1: Part A....................................368
Exercise 5.1: Part B....................................377
5.2 Series of Functions ........................................................................389
Exercise 5.2: Part A....................................395
Exercise 5.2: Part B....................................401
Bibliography 421
Index 423
courseroomsity - Stuvia | Page 3 of 691
, STUDY NOTES & REFERENCE GUIDE
Chapter 1
Metric Spaces
·
In this chapter, the basic concept of metric spaces will be introduced.
Naively, they are simply nonempty sets equipped with a structure
called metric. For the less matured students, at the beginning, this
structure may appear to be a bit abstract and difficult to master.
But in practice, this seemingly new concept is nothing more than
a tiny little abstractization of the familiar space Rn and so all one
needs to do is that whenever one needs to work on a problem in an
abstract metric space, one first looks at the problem on Rn, then one
would be able to see the clue of how to proceed in the general case.
In fact, in general, the most effective way to master a new concept
in any branch of mathematics is to keep in mind a couple of typical
concrete examples and think of these examples all the time. It is just
that easy.
1.1 Definitions and Examples
Definition 1.1.1. Let X be a nonempty set.
A metric on X is a real-valued function
d : X ×X → R
satisf
ying
(M1) d(x, y) ≥ 0 and d(x, y ) = 0 if and only
if x = y,
(M2) (symmetry) d(x, y ) = d(y, x),
(M3) (triangle inequality) d(x, y) ≤ d(x, z)
+ d(z, y)
for all x, y, z ∈ X. Given x, y ∈ X, d(x, y)
is also known as the distance between x and
y with respect to d. The pair (X, d) is
called a metric space and elements in X are
referred to as points in X. For the sake of
convenience, in case there is a clearly
defined metric d on X, we shall simply call
X a metric space.
courseroomsity - Stuvia | Page 4 of 691
