Hilbert Spaces

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Define orthonormal bases for Hilbert spaces, and use them appropriately
• Define and compute the adjoint, operator norm, and spectrum for operators on Hilbert spaces, and derive relations between them
• Define and explain the meaning of infinite sums of vectors, convergence, and absolute convergence in Hilbert spaces
• Derive and explain relations between the convergence of vectors, Cauchy sequences, and the completeness of normed vector spaces
• State and elucidate the spectral theorem for compact self-adjoint operators, and evaluate its potential generalization to other operators
• Define the tensor product of inner product spaces and analyse vectors and operators on the tensor product spaces
• Solve basic-level math problems related to elementary notions in quantum mechanics and quantum information theory
• Define, compute, and analyse the inner products, norms, and convergence of vectors in inner product spaces
• Define linear functionals on inner product spaces, and explain their properties

Part 1 will focus on finite-dimensional Hilbert spaces and matrices:

(1.1) Inner products on vector spaces

(1.2) Norms and convergence of vectors

(1.3) Span of vectors, orthonormal bases, and Parseval’s identity

(1.4) Linear functionals on inner product spaces

(1.5) Adjoint, operator norm, and spectrum for matrices

(1.6) Self-adjoint matrices and normal matrices

(1.7) Unitary matrices and projections

(1.8) Spectral theorem for normal matrices

(1.9) Tensor product of inner product spaces

(1.10) Elementary notions in quantum information theory

Part 2 will focus on infinite-dimensional Hilbert spaces and operators:

(2.1) Completeness of normed vector spaces

(2.2) Definitions and examples of Hilbert spaces and Banach spaces

(2.3) Bounded operators and the operator norm

(2.4) Orthonormal bases for Hilbert spaces

(2.5) Closed subspaces and orthogonal decomposition of Hilbert spaces

(2.6) Linear functionals on Hilbert spaces

(2.7) The Riesz representation theorem

(2.8) Invertibility of operators on Hilbert spaces

(2.9) The Banach isomorphism theorem

(2.10) Spectrum of operators on Hilbert spaces

(2.11) Self-adjoint operators and normal operators

(2.12) Unitary operators and projections

(2.13) Isometries and unilateral shift operators

(2.14) Compact operators

(2.15) Spectral theorem for compact self-adjoint operators

(2.16) Unbounded operators

(2.17) Heisenberg uncertainty principle

Teaching and learning methods

Lectures with exercise problems, tutorials with group work, coursework and private study

Resources & Reading list

Textbooks

Kaye, P., Laflamme, R., & Mosca, M. (2006). An Introduction to Quantum Computing. Oxford University Press.

Young, N. (1988). An introduction to Hilbert space. Cambridge University Press.

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