Real Analysis

  • Summer schools
  • Department of Mathematics
  • Application code SS-ME306
  • Starting 2022
  • Short course: Open
  • Location: Houghton Street, London

Real Analysis is an area of mathematics that was developed to formalise the study of numbers and functions and to investigate important concepts such as limits and continuity. These concepts underpin calculus and its applications. Real Analysis has become an indispensable tool in a number of application areas. In particular, many of its key concepts, such as convergence, compactness and convexity, have become central to economic theory.

This course covers the main aspects of real analysis: convergence of sequences and series and key concepts, including completeness, compactness and continuity, from the particular settings of real numbers and Euclidean spaces to the much more general context of metric spaces.

The course is particularly suitable for students who want to bolster their mathematical background as preparation for postgraduate study in economics and related areas and for professionals who want to follow recent developments in economic theory.

Session: Two
Dates: 11 July - 29 July 2022 
Lecturer: Professor Martin Anthony and Professor Johannes Ruf


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 but may be required for credit by your home institution. Your home institution will be able to advise how you can meet their credit requirements.

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


Courses on multivariate calculus and linear algebra, both at intermediate level. In addition, students need to be familiar with methods of proofs and basic set theory. The course will begin with a brief review of this material. 

Key topics

The course provides a rigorous, but accessible, treatment of real analysis and analysis on metric spaces and will be delivered by formal lectures supported by interactive classes.

  • Basics: proof, logic, sets and functions
  • Real numbers and sequences
  • Functions, limits and continuity
  • Infinite series
  • Metric and normed spaces
  • Convergence, completeness and compactness
  • Continuity in metric spaces
  • The derivative
  • Convexity
  • Fixed point theorems

Programme structure and assessment

This course is delivered via a combination of lectures and classes.

The course is assessed through two examinations: one mid-session examination (50%) and one final examination (50%). Take-home exercises are assigned for class preparation, and students will work on additional exercises during classes. None of these exercises will count towards the final grade. They are designed as formative assessments to build intuition and understanding and to help students prepare for the summative assessments.

Course outcomes

After completing this course, students will:

  • Have gained a thorough grounding in modern Real Analysis, including key concepts such as convergence, continuity, completeness and compactness
  • Be able to comprehend and critically reflect on mathematical statements and their proofs and to write their own formal proofs of mathematical results
  • Have developed a higher capacity for abstract and rigorous mathematical reasoning
  • Be well-equipped to study advanced applications of Real Analysis in disciplines such as Economics

Is this course right for you?

This course will suit you if you want to develop your mathematical skills. You will learn how to do rigorous proofs and you will understand the key concepts of Real Analysis. These concepts play a central role in Economic Theory and in other domains. 

Your department

LSE’s 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 has more than doubled in size in recent years, and this growth trajectory reflects the increasing impact that mathematical theory and techniques are having on subjects such as economics, finance and many other areas of the social sciences.

Students will engage with world-leading faculty and be exposed to cutting-edge research in the field, at the forefront of the intersection between mathematics and its use in other social science disciplines to solve global problems. This ensures that students within the department are equipped with the necessary analytical skills to tackle important mathematical challenges in a variety of sectors.

Your faculty

Professor Martin Anthony
Professor of Mathematics, Department of Mathematics

Professor Johannes Ruf
Professor of Mathematics, Department of Mathematics

Reading materials

A comprehensive set of lecture notes (with exercises) will be provided. There is no set text, but useful additional reading is provided by:
Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis. Wiley, 2011


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