The first half of this module (Functional Analysis) introduces Hilbert spaces, Banach spaces, dual spaces and linear operators, and explores the interaction between linear algebra and analysis in the study of infinite dimensional spaces. The second half (Fourier Analysis) introduces and develops the classical theory of Fourier series and the Fourier transform on the real line, with emphasis on both mathematical rigour and applications.

Learning Outcomes

By the end of the module students should be able to:

Work in infinite dimensional Hilbert and Banach spaces, find bases for Hilbert spaces, and use several different notions of convergence.

Combine ideas from algebra and analysis to solve problems in functional analysis.

Understand the basic theory of Fourier series, the Fourier transform, and other related transforms. This will include criteria for the convergence of Fourier series and the invertibility of the Fourier transform.

Use Fourier analysis as a tool to understand important classes of partial differential equations, and describe further applications within mathematics.

Assessment

31129-01 : Raw Module Mark : Coursework (100%)

Assessment Methods & Exceptions

3 hour Written Unseen Examination (80%); In-course Assessment (20%).