Hilbert Spaces

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Determine whether linear operators are continuous, invertible, self-adjoint, compact etc, and determine adjoints
• Define Banach and Hilbert spaces and be familiar with various examples of these
• State and apply the Banach Isomorphism Theorem and Closed Graph Theorem to determine whether operators are bounded
• Determine whether a subset of a normed space is complete or compact
• 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

• 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..

Teaching and learning methods

Lectures, tutorials, problem sheets and private study

Resources & Reading list

Textbooks

S Berberian. Introduction to Hilbert space.

N Aronszajn. Introduction to theory of Hilbert spaces.

N Young. An introduction to Hilbert space.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat Information

Repeat type: Internal & External