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## MATH3215 Hilbert Spaces and Fourier Analysis

### 15 creditsClass Size: 30

Module manager: Dr Derek Harland
Email: D.G.Harland@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2017/18

### Pre-requisite qualifications

MATH2016, or equivalent. MATH3210 is useful, but not required.

This module is approved as a discovery module

### Module summary

A Hilbert space is an infinite-dimensional analogue of Euclidean space. It possesses the structure of a vector space together with notions of orthogonality and distance. The techniques of Fourier series resolve complicated wave forms into frequencies and phases. From those frequencies and phases the original wave form can be recaptured. In the last third of the module, we shall develop the theory of linear operators, which are the natural generalization of matrices and mappings on finite-dimensional spaces. We shall examine the role eigenvalues play in this setting and see how they generalise to the notion of spectrum.

### Objectives

On completion of this module, students should be able to:
a) calculate the Fourier coefficients of certain elementary functions;
b) perform a range of calculations involving orthogonal expansions in Hilbert spaces;
c) apply functional analytic techniques to the study of Fourier series;
d) give the definitions and basic properties of various classes of operators (including the classes of compact, Hermitian, and unitary operators) on a Hilbert space, and use them in specific examples;
e) prove results related to the theorems in the course.

### Syllabus

1. Real and complex Fourier series. Vibrating string.
2. Banach spaces (basic definitions only).
3. Hilbert spaces. Subspaces. Linear spans. Orthogonal expansions. Bessel's inequality. The Riesz-Fischer theorem. Orthogonal complements.
4. Applications to Fourier series. Fejer's theorem. Parseval's formula. Sums of numerical series. The Weierstrass approximation theorem.
5. Dual space of a normed space. Self-duality of Hilbert space.
6. Linear operators. Adjoint. Hermitian, unitary and normal operators. B(H) as a Banach space. The spectrum of an operator on a Hilbert space. The spectral radius formula.
7. Compact operators. Hilbert-Schmidt operators.
8. The spectral theorem for compact Hermitian operators.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Private study

- Reading lecture notes: 78 hours
- Solving coursework problems: 25 hours
- Preparing for examinations: 15 hours

### Opportunities for Formative Feedback

Regular exercise sheets.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated