# MMath Mathematics with Financial Mathematics

Year of entry: 2023

## Course unit details:Linear Analysis

Unit code MATH31002 10 Level 3 Semester 2 No

### Overview

This course unit deals with a coherent and elegant collection of results in analysis. The aim of this course is to provide an introduction to the theory of infinite dimensional linear spaces, which is not only an important tool, but is also a central topic in modern mathematics.

This area has many applications to other areas in Pure and Applied Mathematics such as Dynamical Systems, C* algebras, Quantum Physics, Numerical Analysis, etc.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Real Analysis A MATH20101 Pre-Requisite Compulsory
Real Analysis B MATH20111 Pre-Requisite Compulsory
Metric Spaces MATH20122 Pre-Requisite Compulsory

Students must have taken MATH20122 and (MATH20101 OR MATH20111)

### Aims

To give an introduction to Modern Analysis, including elements of functional analysis. The emphasis will be on ideas and results.

### Learning outcomes

On successful completion of this module students will be able to:

• prove the Weierstrass Approximation Theorem and apply the Stone-Weierstrass Theorem to the approximation of continuous functions;
• define Hilbert and Banach spaces, with C[0,1]and the l^p spaces serving as examples;
• define continuous linear functionals on Banach spaces and dual spaces, prove the Riesz representation theorem for Hilbert spaces and calculate the norms of linear functionals on lp spaces;
• define continuous linear operators, their spectrum (with matrices, the diagonal operator and the shift operator serving as examples) and their spectral radius;
• define self-adjoint, isometric, unitary and compact operators on Hilbert spaces and prove the structural theorem on their spectra.

Future topics requiring this course unit

This course would be useful to students interested in the following topics: MATH41012 Fourier Analysis and Lebesgue Integration.

### Syllabus

1.Revision of compactness and uniform continuity. [2 lectures]

2.Approximation by polynomials and the Stone-Weierstrass Theorem. 

3.Normed vector spaces. Finite and infinite dimensional spaces. Completeness and Banach spaces. 

4.Sequence spaces. Continuous function spaces. Separability. Hilbert spaces, orthogonal complements and direct sum decompositions, orthonormal bases. 

5.Bounded linear functionals. The Hahn-Banach theorem. Dual spaces. The Riesz representation theorem for Hilbert spaces. 

6.Bounded linear operators and their norms. Compact operators. Invertible operators. Spectra. 

7.Linear operators on Hilbert spaces. Self-adjoint and unitary operators and their spectra. 

### Assessment methods

End of semester examination: two hours weighting 100%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• G. F. Simmons, An Introduction to Topology and Modern Analysis, McGraw Hill, 1963.
• I.J. Maddox, Elements of Functional Analysis, C.U.P. 1989.
• B.P. Rynne and M.A. Youngson, Linear Functional Analysis, Springer S.U.M.S., 2008.

### Study hours

Scheduled activity hours
Lectures 22
Tutorials 11
Independent study hours
Independent study 67

### Teaching staff

Staff member Role
Nikita Sidorov Unit coordinator