Real Analysis

Real analysis is the area of mathematics dealing with real numbers and the analytic properties of real-valued functions and sequences. In this course we shall develop concepts such as convergence, continuity, completeness, compactness and convexity in the settings of real numbers, Euclidean spaces, and more general metric spaces.

Real analysis is part of the foundation for further study in mathematics as well as graduate studies in economics. A considerable part of economic theory is difficult to follow without a strong background in real analysis. For example, the concepts of compactness and convexity play an important role in optimisation theory and thus in microeconomics.   

This course in real analysis is designed to meet the needs of economics students who are planning to study at postgraduate level. It is also suitable for mathematics and statistics students. 

Course Benefits
After completing this course students will:

• gain knowledge of concepts of modern analysis, such as convergence, continuity, completeness, compactness and convexity in the setting of Euclidean spaces and more general metric spaces;
• develop a higher level of mathematical maturity combined with the ability to think analytically
• be able to write simple proofs on their own and study rigorous proofs
• be able to follow more advanced treatments of real analysis and study its applications in disciplines such as economics.

Prerequisites
A course on multivariate calculus and linear algebra, both at first year undergraduate level. Furthermore, students need to be familiar with methods of proofs, basic set theory and the properties of real numbers. A revision of this material will be covered in the foundations part of the course and can also be found in Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert (2011) [Chapters 1 and 2 and Appendix A].

This course has two components and will be delivered online and on-campus.

Foundation (10-14 August 2015)
This part of the course will be delivered online, and will consist of:

• Video lectures, delivered in short segments
• Lecture notes, exercises and solutions.

On the Friday of this week, students will be asked to complete an online diagnostic quiz. Although mandatory, the quiz is not an exam, and passing it is not a prerequisite of attending the course. It is instead intended to give the lecturer an idea of what will need to be looked at on the first afternoon campus session. 

Core (17-28 August 2015)
To be delivered on campus over 9 lectures and 8 classes. Most lectures will involve the use of Class Response System "clickers" to help students think about the concepts and clarify the material covered before attempting exercises.

Assessment: A set of homework exercises (15%) and a two hour comprehensive final exam (85%). The final examination will take place on the afternoon of Friday 28 August 2015.

The course provides a rigorous yet accessible treatment of real analysis and analysis on metric spaces and will be delivered by formal lectures supported by classes. Students are required to solve exercises (homework assignments) on a daily basis. Exercises are a vital part of the course and their solutions will be discussed in the classes.

Part I: Foundations

• Preliminaries: revision of logic, methods of proof, set theory, and properties of real numbers
• Sequences in ℝ
• Infinite series in ℝ

Part II: Core

• Metric spaces
• Continuity in ℝ and in metric spaces
• Differentiation in ℝ and in Euclidean spaces
• Fixed point theorems
• Convex functions and sets
• Separation theorems
• Applications

Main Texts
There is no set text for this course. A comprehensive course pack with course notes and exercises will be supplied.

Suggested references
Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert (2011)
Introduction to Metric and Topological Spaces by W. A. Sutherland (1995)
Principles of Mathematical Analysis by W. Rudin (1976)
The Elements of Real Analysis, by Robert G. Bartle, 2nd edition (1976)
Convex Functions by A. Wayne Robers and Dale E. Varberg. (1973)

The following book is a useful, more advanced complementary reading:
Real Analysis with Economic Applications by Efe A. Ok (2007).

"It's a great programme, and the Real Analysis course is excellent. Eleni truly cares about her students' progress and her teaching style makes a potentially dry subject very entertaining and intuitive."
2014 participant on Real Analysis

"I enrolled in the course Real Analysis because it is very important for an economist but not offered in my university. LSE gave me an opportunity to fill in the gap and I got more than what I came for.  The course was hard but still understandable, and it has given me a clear idea of how analysis works. The teaching method was excellent and the interaction between teachers and students was better than any of my previous experiences."
2013 participant on Real Analysis