Mathematics

Real Analysis

Module code: G5145
Level 5
15 credits in spring semester
Teaching method: Lecture
Assessment modes: Coursework, Unseen examination

Real Analysis serves as a gentle introduction to the exciting theory of metric spaces: a fundamental theory in mathematical analysis and a prerequisite for several modules in Years 3 and 4.

A metric space is a set together with a function, called a metric, which measures the distance between any two elements of the set. For example, the absolute value of the difference between two real numbers is a metric, turning the set of reals into a metric space. This simple structure – a set and a distance – encompasses many important examples. It enables us to study notions familiar from the real numbers, such as convergence of sequences or continuity of functions, in a unified framework. This gives rise to a rich and elegant theory for metric spaces which recovers many known properties of the real numbers, but also provides new results for more complex examples, like sets of sequences or sets of functions, with vast applicability.

As an application, the module focuses on functions defined on the real line. It studies power series, uniform convergence, and introduces Lebesgue spaces, an important space of functions for mathematical analysis as well as subsequent modules.

Module learning outcomes

• Appreciate rigorous arguments about sequences and series of functions and be able to deploy them in solving problems in analysis;
• Understand the concepts and definitions of pointwise and uniform convergence of sequences of functions;
• Understand the concepts and definitions of metric spaces, separability and completeness, provide and explain examples and counterexamples.