ME406 Real Analysis

The 2016 Methods Summer Programme is now closed. 
Details for 2017 programme will be available soon.

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 LSE Methods Summer Programme course in real analysis is designed to meet the needs of economics students who are planning to study at postgraduate level as well as professionals who need to follow developments in economic analysis and research.

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 
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 
To be delivered on campus over 9 lectures and classes. The course delivery follows the Round Table model and engages students in daily learning activities that enable them to master the mathematical concepts and techniques covered in the course.

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 26 August 2016.

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. 

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)
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).

"This course really provides you with the opportunity to gain a strong background with which you can build on. It gives you the overall picture much better than any of the other maths course that I have done."
2014 participant on Real Analysis

"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

Please note: A full timetable will be provided at registration on Monday 15 August. The expected timetables below contain approximate hours only.

	Week One (hours)
	M	Tu	W	Th	F
Morning lecture	3	3	3	3	3
Afternoon class	1.5	1.5	1.5	1.5	1.5

	Week Two (hours)
	M	Tu	W	Th	F
Morning lecture	3	3	3	3	/
Afternoon class	1.5	1.5	1.5	1.5	Exam