Hilbert Spaces

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Use inner product and Hilbert space structure (Cauchy–Schwarz, orthogonality, projection theorem) to solve best-approximation problems and analyse norm convergence of sequences and series.
• Produce coherent mathematical exposition with explicit assumptions and scope (including spaces/domains) and logical structure; diagnose gaps or incorrect claims by auditing assumption-dependence; and construct corrected, robust arguments with documented checks (e.g. counterexamples, special cases, alternative derivations), including when working with AI-assisted material.
• Transfer Hilbert space methods to applied contexts, including Fourier/wavelets, weak PDE/Galerkin formulations, RKHS/kernels, probability (martingales), and quantum mathematical structure.
• Explain completeness and completion; use orthonormal systems and bases; relate Fourier expansions and Bessel/Parseval identities to L2 convergence and approximation error bounds.
• Use bounded-operator language (operator norm, adjoint, spectrum/resolvent) and the Riesz representation theorem, and distinguish key infinite-dimensional phenomena (spectrum vs eigenvalues; bounded vs unbounded operators; domain and density issues).

Weeks 1–2: Geometry and approximation in Hilbert spaces: real and complex inner products; induced norm; orthogonality and Cauchy–Schwarz; completeness and definition of Hilbert space; canonical models l2 and L2; closed subspaces and orthogonal complements; projection theorem and best approximation; orthonormal systems and bases; Bessel and Parseval identities; Fourier series as prototype of orthonormal expansions. Weeks 3–5: Operators, spectrum, and composition: bounded linear operators and operator norm; linear functionals and Riesz representation; adjoints; self-adjoint and normal operators; compact operators and finite-rank approximation; spectrum and resolvent; spectral theorem for compact self-adjoint operators; orthonormal eigenbases and diagonalisation; tensor products; pure tensors and induced inner product; Schmidt/SVD viewpoint; composite systems and links to kernels, probability, and quantum models. Weeks 6–11: Hilbert space methods in context: Fourier expansions in L2; Parseval/Plancherel; derivative as unbounded operator; Sobolev space H1; weak derivatives and boundary conditions; Lax–Milgram theorem; existence, uniqueness, stability; Galerkin method and orthogonality-based error bounds; positive definite kernels and reproducing kernel Hilbert spaces; representer theorem and regularisation; random variables as L2 elements; conditional expectation as orthogonal projection; L2 convergence; filtrations and martingales; orthogonal increments and Itô isometry; quantum model in L2(R); self-adjoint observables; Fourier transform as unitary; Robertson–Schrödinger and Heisenberg uncertainty; Gaussian equality cases.

Lectures with exercise problems, tutorials, coursework and private study

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

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.