MATH4001

MATH4001: Functional Analysis

Science and Engineering

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About this Unit

This unit covers advanced topics in Functional Analysis and Advanced Calculus which are important in several areas of applied mathematics, physics, and engineering. The first part of this unit deals with functional spaces, which are infinite-dimensional vector spaces whose elements are, as the name suggests, functions. To develop the adequate background for the study of these spaces, we will introduce the notions of normed and metric spaces, Banach and Hilbert spaces. These have immense relevance in contexts like applied optimisation and quantum physics. We will then introduce the theory of generalised functions, also known as distributions. Distributions are routinely employed in science and engineering to model impulsive behaviours: the classical example is the Dirac delta distribution, which is used to model, for example, a point charge, point mass or electron point. Enlarging the classical Lebesgue functional spaces to include distributions allows us to define the appropriate set of solutions of linear partial differential equations. To address this topic, which is one of the pivotal ones in modern theoretical and applied mathematics, we will introduce and develop the theory of the Fourier transform, not only for functions, but also for tempered distributions, and we will apply it to the solution of important partial differential equations arising in mathematical physics, including the heat equation, the Laplace equation and the Schroedinger equation.

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Unit Details

Credits25

Contact Hours3 Weekly

Result TypeGrade/Mark

AreaSchool of Electrical Engineering, Computing and Mathematical Sciences (EECMS) (1153)

Field of Education010101 Mathematics

Handbook View Official Handbook Page →

Tuition Pattern

Lecture1 x 2 hours weekly
Seminar1 x 1 hour weekly

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