Linear Analysis (20cr)

Code

MATH4090

School

Mathematical Sciences

Level

4

Credits

20

Semesters

Spring UK

Summary

This course gives an introduction to some basic ideas of functional analysis and Lebesgue measure and Lebesgue integration. Many concepts from linear algebra in finite dimensional vector spaces (e.g., writing a vector in terms of a basis, eigenvalues of a linear map, diagonalisation etc.) have generalisations in the setting of infinite dimensional spaces making this theory a powerful tool with many applications in pure and applied mathematics. The theory of Lebesgue integral is an important tool in analysis, which is a generalisation of Riemann integral. 

•    Normed, normed linear spaces, completeness, separable normed linear spaces, Banach spaces, linear subspaces, basic examples.

•    Lebesgue outer measure, Lebesgue measure, Lebesgue measurable functions and Lebesgue integral in R^n; monotone convergence theorem and the Dominated convergence theorem, completeness of L^p spaces.

•    Inner product spaces, Hilbert space, orthonormal systems and orthonormal basis, Fourier series with respect to orthonormal systems, Bessel's inequality, Parseval's identity, separable Hilbert space. orthogonal projections, l^2 as main example.

•    Linear operators and the operator norm; bounded linear operators, composition of bounded linear operators, invertible bounded linear operators, bounded linear functionals, dual spaces, completeness of spaces of bounded linear operators, the shift operator, the multiplication operator, Riesz-Fréchet representation theorem.
 

Target Students

Single and Joint Honours students from the School of Mathematical Sciences. Mathematical Physics students.

Classes

One two-hour class and two one-hour classes per week timetabled centrally, some of which may be used for lectures or workshops.

Assessment

• 10% Coursework 1: Written Coursework.
• 90% Written Exam 1 (3-hour): Written exam in person.

Assessed by end of spring semester

Educational Aims

Learning Outcomes

L1- state the basic theorems of integration theory and Functional Analysis;

L2- define and apply the main concepts of functional analysis.

L3- prove basic propositions about integration theory, normed linear spaces, and operators on normed linear spaces;

L4- apply methods and concepts of functional analysis in the investigation of examples;

Conveners

• Dr Joel Feinstein