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.
Pre-requisites: (MATH1049 AND MATH2039 AND MATH2049) OR (MATH1049 AND MATH2039 AND MATH2046)
Aims and Objectives
Having successfully completed this module you will be able to:
- Define the spectrum of an operator, and derive basic properties
- Apply the theory of operators on Hilbert space to differential operators
- 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 Banach and Hilbert spaces and be familiar with various examples of these
- Determine the index of an operator in simple cases and derive basic properties
- Determine whether a subset of a normed space is complete or compact
- 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=Δ+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
|Total study time||150|
Resources & Reading list
N Aronszajn. Introduction to theory of Hilbert spaces.
N Young. An introduction to Hilbert space.
S Berberian. Introduction to Hilbert space.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External