The University of Southampton
Courses

# MATH3076 Hilbert Spaces

## Module Overview

This module is an introduction to functional analysis on Hilbert spaces. The subject of functional analysis builds on the linear algebra studied in the first year and the analysis studied in the second year. The module introduces the concept of Hilbert spaces - an infinite dimensional analogue of Euclidean space. The definition of a vector space only allows the construction of finite sums of vectors. By introducing the norm of a vector (i.e. the length of a vector), it is then possible to do analysis with vectors, compute limits, take infinite sums etc. In mathematics, the study of Hilbert spaces has applications including Fourier series, the Fourier transform, and the solution of differential equations. Hilbert spaces also play an important role in physics, forming the mathematical basis for Quantum Mechanics The module builds up to classical results such as the Toeplitz index theorem, the spectral theory of compact operators, and applications to differential operators.

### Aims and Objectives

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Define Banach and Hilbert spaces and be familiar with various examples of these
• Determine whether a subset of a normed space is complete or compact
• Determine whether linear operators are continuous, invertible, self-adjoint, compact etc, and determine adjoints
• State and apply the Banach Isomorphism Theorem and Closed Graph Theorem to determine whether operators are bounded
• Define the spectrum of an operator, and derive basic properties
• Determine the index of an operator in simple cases and derive basic properties
• Apply the theory of operators on Hilbert space to differential operators

### Syllabus

• Norms and inner products on vector spaces. Sequences and limits in normed spaces. • Completeness and definition of Hilbert space. Example of l 2. Example of L 2[0,1] as the completion of C[0,1]. Brief introduction to measure and Lebesgue integration. • Sequential compactness and applications. Weak convergence. • The Projection Theorem. Riesz representation theorem (for Hilbert spaces). • Orthonormal bases and Gram-Schmidt process. Isomorphism of separable spaces with l 2. • Example of Fourier series. • Bounded operators and the operator norm. Adjoints. • Invertibility of operators and the Banach Isomorphism Theorem. • Self-adjoint/unitary/normal operators. • Spectrum and approximate eigenvalues. • (If time) Fourier transform: Plancherel theorem and Fourier inversion. • Diagonalisation. Compact operators. Spectral Theorem (compact self-adjoint case). • Fredholm operators. The index zero criterion and Atkinson’s parametrix criterion. • Spectral theory for differential operators of the form D=?+V(x): Rellich's Compactness Lemma. Differential operators on Sobolev spaces. Elliptic regularity and Weyl's lemma..

### Learning and Teaching

#### Teaching and learning methods

Lectures, tutorials, problem sheets and private study

TypeHours
Independent Study108
Teaching42
Total study time150

N Young. An introduction to Hilbert space.

N Aronszajn. Introduction to theory of Hilbert spaces.

S Berberian. Introduction to Hilbert space.

### Assessment

#### Summative

MethodPercentage contribution
Class Test 10%
Coursework 10%
Exam 80%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

#### Pre-requisites

To study this module, you will need to have studied the following module(s):

CodeModule
MATH1049Linear Algebra II
MATH2039Analysis

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.