Real Analysis

  • Summer schools
  • Department of Mathematics
  • Application code SS-ME306
  • Starting 2021
  • Short course: Closed
  • Location: Houghton Street, London

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 as well as professionals who need to follow developments in economic analysis and research.

Session: Two
Dates: 12 – 30 July 2021 
Lecturer: Dr Eleni Katirtzoglou


Programme details

Key facts

Level: 300 level. Read more information on levels in our FAQs

Fees:  Please see Fees and payments

Lectures: 36 hours 

Classes: 18 hours

Assessment*: A midession exam during the second week of the course and a comprehensive final exam on the Friday of the third week.

Typical credit**: 3-4 credits (US) 7.5 ECTS points (EU)

*Assessment is optional

**You will need to check with your home institution

For more information on exams and credit, read Teaching and assessment


A course on multivariate calculus and linear algebra, both at intermediate level. Furthermore, students need to be familiar with methods of proofs, basic set theory and the properties of real numbers. A revision of methods of proofs and basic set theory will be covered at the start 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]. For more information on the background required see here.

Programme structure

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. The delivery of the course is based on the Round Table model for which Dr Katirtzoglou won the LSESU Teaching Excellence Award for Innovative Teaching. 

  • Preliminaries: revision of logic, methods of proof, set theory, and properties of real numbers
  • Sequences in ℝ
  • Infinite series in ℝ
  • Metric spaces
  • Continuity in ℝ and in metric spaces
  • Differentiation in ℝ and in Euclidean spaces
  • Fixed point theorems
  • Convex functions and sets (if time allows)
  • Separation theorems (if time allows)

Course outcomes

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.


The LSE Department of Mathematics is internationally recognised for its teaching and research. Located within a world-class social science institution, the department aims to be a leading centre for Mathematics in the Social Sciences. The Department of Mathematics was submitted jointly to REF 2014 with LSE's Department of Statistics: 84% of the research outputs of the two departments were classed as either world-leading or internationally excellent in terms of originality, significance and rigour.

The Department has more than doubled in size over the past few years, and this growth trajectory reflects the increasing impact that mathematical theory and mathematical techniques are having on subjects such as economics and finance, and on many other areas of the Social Sciences.

On this three week intensive programme, you will engage with and learn from full-time lecturers from the LSE’s mathematics faculty.

Reading materials

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).

**Course content, faculty and dates may be subject to change without prior notice

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