MS-E1461: Hilbert Spaces
| Course name | Hilbert Spaces |
|---|---|
| Course code | MS-E1461 |
| Abbreviation | Hilbert |
| Period | I |
| Lecturer | Ville Turunen |
Description
As the name suggests, the main focus is on Hilbert spaces, which are especially relevant in quantum mechanics, partial differential equations, Fourier analysis, and spectral theory.
Additionally, the course offers basic methods that are helpful in other master’s level mathematics courses as well.
Learning outcomes official
You will learn about norms and inner products in infinite-dimensional vector spaces. Related to these structures, you will understand basic properties of bounded linear operators and duality in Hilbert spaces, together with diagonalization of compact self-adjoint operators.
Content official
Orthogonality, orthonormal bases, bounded linear operators, functionals, and elementary spectral theory in Hilbert spaces.
Topics include the Jordan–von Neumann theorem, the Riesz representation theorem for Hilbert spaces, diagonalization of compact self-adjoint operators, the Hilbert–Schmidt spectral theorem, and singular value decomposition.
Course material
Official material
There are quite good lecture notes, available as PDF written in LaTeX, which are enough for the course. Additionally, lecture videos from 2021 are available and cover almost the same topics as the actual lectures.
Extra material
Contents and workload
As with other standard master’s level mathematics courses, this one requires a lot of work. The workload is fairly balanced between weeks.
Overall workload
Weekly contents
| Week | Topics |
|---|---|
| 1 | Inner products, Hilbert spaces, polarization identity, $\ell^p$ spaces, bounded and continuous linear operators, extension of bounded operators. |
| 2 | Direct sums of Hilbert spaces, orthogonality, Hilbert integral inequality, orthogonal complements, closed subspaces. |
| 3 | Pythagorean equality, Bessel’s inequality, orthonormal collections, Gram–Schmidt process, countable orthonormal bases, and separability. |
| 4 | Bounded operators on Hilbert spaces, adjoint operators, self-adjointness, operator norm identities, Fréchet–Riesz theorem, and Hahn–Banach type extension. |
| 5 | Hermitian forms, Banach–Steinhaus theorem, Lax–Milgram theorem, normal, self-adjoint, unitary, positive, and compact operators. |
| 6 | Shift operators, adjoints, spectra, compact self-adjoint operators, Hilbert–Schmidt spectral theorem, spectral decompositions, and polar decompositions. |
Practicalities
The course can be graded in two ways:
- Weekly exercises 1/3 and an exam 2/3.
- Exam 100%.
Related courses
Official prerequisites
MS-A00XX, MS-A01XX, MS-C1540
Additional prerequisites
Metric Spaces and Linear Algebra is recommended.
More like this
MS-E1462: Banach Spaces
Trivia
Timestamp
Based on the 2025 version of the course, taught in period I.