Analysis II
Lecturer: Neshan Wickramasekera
Michaelmas Term 2015
These notes are produced entirely from the course I took, and my subsequent thoughts.
They are not necessarily an accurate representation of what was presented, and may have
in places been substantially edited.
,Analysis II
Contents
Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1. Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Properties of Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2. Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3. Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2. Uniform Continuity and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1. The Weierstrass Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2. Application: Riemann Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3. Non-Examinable: Lebesgue’s Theorem on the Riemann Integral . . . . . . . . . . . 21
3. !n as a Normed Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1. Equivalence of Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2. Completeness in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3. Open and Closed sets in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4. Sequentially Compact Subsets of Normed Spaces . . . . . . . . . . . . . . . . . . . . 37
3.5. Mappings between Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4. Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1. Completeness in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2. Sequential Compactness in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3. Continuous Mappings between Metric Spaces . . . . . . . . . . . . . . . . . . . . . . 52
5. The Contraction Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1. Application: Existence Theorem for ODE’s . . . . . . . . . . . . . . . . . . . . . . . . 56
6. Differentiation in !n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.1. Consequences of Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2. The Operator Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
1
,Analysis II
6.3. Mean Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.4. The Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.5. 2nd Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2
, Analysis II
INTRODUCTION AND MOTIVATION
In this course we shall further develop the tools seen in Analysis I. We start by studying the real-
valued functions and look at different notions of convergence for them. This will lead us to working
out when we can swap limits with the integral sign and derivatives. Then we look at !n as a key
example of finite-dimensional vector spaces, and show that it the properties of !n are independent
of the choice of norm we put on it. We shall then generalise to metric spaces and have a brief detour
into topology and how the added generality can help us. We will then prove a major result, the
Picard-Lindelöf theorem, which tells us when we can solve certain types of ODEs. We then finish by
looking at looking at differentiation in dimensions > 1.
Differential equations (and more generally partial differential equations) are of fundamental impor-
tance in theoretical physics and so understanding when we are able to solve them is crucial. To
do this we will need to understand what it means for functions to converge, and choose the right
meaning of convergence so that it is actually useful.
3
Lecturer: Neshan Wickramasekera
Michaelmas Term 2015
These notes are produced entirely from the course I took, and my subsequent thoughts.
They are not necessarily an accurate representation of what was presented, and may have
in places been substantially edited.
,Analysis II
Contents
Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1. Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Properties of Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2. Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3. Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2. Uniform Continuity and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1. The Weierstrass Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2. Application: Riemann Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3. Non-Examinable: Lebesgue’s Theorem on the Riemann Integral . . . . . . . . . . . 21
3. !n as a Normed Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1. Equivalence of Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2. Completeness in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3. Open and Closed sets in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4. Sequentially Compact Subsets of Normed Spaces . . . . . . . . . . . . . . . . . . . . 37
3.5. Mappings between Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4. Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1. Completeness in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2. Sequential Compactness in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3. Continuous Mappings between Metric Spaces . . . . . . . . . . . . . . . . . . . . . . 52
5. The Contraction Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1. Application: Existence Theorem for ODE’s . . . . . . . . . . . . . . . . . . . . . . . . 56
6. Differentiation in !n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.1. Consequences of Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2. The Operator Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
1
,Analysis II
6.3. Mean Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.4. The Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.5. 2nd Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2
, Analysis II
INTRODUCTION AND MOTIVATION
In this course we shall further develop the tools seen in Analysis I. We start by studying the real-
valued functions and look at different notions of convergence for them. This will lead us to working
out when we can swap limits with the integral sign and derivatives. Then we look at !n as a key
example of finite-dimensional vector spaces, and show that it the properties of !n are independent
of the choice of norm we put on it. We shall then generalise to metric spaces and have a brief detour
into topology and how the added generality can help us. We will then prove a major result, the
Picard-Lindelöf theorem, which tells us when we can solve certain types of ODEs. We then finish by
looking at looking at differentiation in dimensions > 1.
Differential equations (and more generally partial differential equations) are of fundamental impor-
tance in theoretical physics and so understanding when we are able to solve them is crucial. To
do this we will need to understand what it means for functions to converge, and choose the right
meaning of convergence so that it is actually useful.
3
