This course focuses on Hilbert spaces and Fourier analysis, with applications to classical linear partial differential equations (PDEs).
This course will cover the following topics
You can find all the relevant informations about the course in the course catalog.
Lectures take place in HG E 1.2 on Thursdays from 13:15 to 14:00 and on Fridays in HG G 3 from 10:15 to 12:00.
Exercise classes take place on Wednesday in different locations and different groups according to the table below. Please enroll to the exercise classes in MyStudies. The first exercise class will be on 25 February 2026. The last exercise class will be on 27 May 2026.
| When | Where | Who | Language |
|---|---|---|---|
| Wednesday 10:15-12:00 | HG E 33.5 | Tom Julius Becker | English or German |
| Wednesday 10:15-12:00 | LEE D 105 | Chiara Polito | English |
| Wednesday 10:15-12:00 | ML F 40 | Johannes Luber | English or German |
| Wednesday 10:15-12:00 | ML J 34.1 | Derk Steffens | German |
| Wednesday 12:15-14:00 | LFV E 41 | Lluís Marill Farré | English or German |
Currently, there are no office hours, but they will be established later if necessary. If you wish to schedule a meeting, please contact the coordinator directly at mdimarco@ethz.ch for an appointment.
Please note that the numbering of the theorems, exercises, etc., as well as the page numbers in the diary, refers to the current version of the notes.
| Date | Topic discussed | Notes |
|---|---|---|
| Thursday, February 19 - Friday, February 20 | Introduction to the course, definition of (real and complex) vector spaces and inner product spaces. Examples: \(\mathbb{R}^d\) , \(\mathbb{C}^d\), \(\ell^2(\mathbb{C})\). Examples of inner product spaces \(L^2(0,1)\), \(C([0,1])\) (examples 1.8 and 1.11). Norms, Cauchy-Schwarz inequality and triangular inequality with proof, Parallelogram identity, Pythagora’s theorem, Polarisation formulas, Ptolemy inequality. Lemma 1.25 (continuity of the inner product), exercise 1.2, example 1.28 (some norms on finite dimensional), example 1.29 (\(C([0,1])\) is an infinite dimensional vector space) and some classical norms. | Pages 1-16 |
| Thursday, February 26 | \(L^p\) spaces are normed spaces. Some topological definitions: open ball, interior point, open set, closed set, convex set, topological vector space, Cauchy sequences, completeness. Comparison between norms on the same space. | Pages 16-20 |
| Friday, February 27 | Definition of Hilbert spaces, examples of Hilbert spaces (\(\mathbb{C}^d\), \(\ell^2\), \(L^2\)); a subspace of a Hilbert space is Hilbert if and only if it is closed; example of inner product space that is not complete \((C([-1,1]),\|\cdot\|_{L^2})\), notion of density and closure of a set, separability, example of separable spaces. Bessel inequality and Parseval identity (Theorem 1.52). Definition of Hilbert basis. Statement of the completeness criterion for separable Hilbert Spaces (Theorem 1.54). | Pages 20-28 |
| Thursday, March 5 | Proof of the completeness criterion for separable Hilbert Spaces (Theorem 1.54); existence of an Hilbert basis (Theorem 1.55); standard basis of \(\ell^2\) (Example 1.56); example of compactly supported sequences (Example 1.57). | Pages 28-30 |
| Friday, March 6 | Every separable Hilbert space of infinite dimension is isometric to \(\ell^2\) (Corollary 1.58). Existence of the projection on closed subspaces and characterization of the orthogonal projection (Theorem 1.59). Projection on closed convex sets (Remark 1.60). Nontriviality of the orthogonal space of a closed proper subspace (Corollary 1.61). Definition of orthogonal complement and its properties (Definition 1.65, Remarks 1.66, 1.67, 1.68). The orthogonal decomposition is an isometry of Hilbert spaces (Proposition 1.69). | Pages 30-36 |
| Thursday, March 12 - Friday, March 13 | Definition of linear/bounded/unbounded operators; example of an unbounded operator (derivative); definition of the norm of a linear operator and proof that it is equal to the Lipschitz constant; continuity of linear operators if the dimension is finite; example: the identity is not necessarily continuous if we change the topology; a linear operator between normed vector spaces is continuous if and only if it is bounded. Riesz' representation Theorem of continuous linear functionals on an Hilbert space. Radon-Nikodym Theorem (without proof). | Pages 36-41 |
| Thursday, March 19 | Examples of applications of Riesz Representation Theorem. Overview of the Axiom of Choice (not examinable). | Pages 41-48 |
| Friday, March 20 | Motivation of the Fourier series, definiton of Fourier coefficients and Fourier partial sum. The Complex Stone Weierstrass Theorem (only statement). Theorem: the Fourier basis forms an Hilbert basis of \(L^2((-\pi,\pi);\mathbb{C})\) (proof using the Complex Stone Weierstrass). Fourier series do not uniformly approximate periodic functions. | Pages 49-54 |
| Thursday, March 26 | Consequences of the fact that the Fourier basis is an Hilbert basis: \(L^2\) convergence of partial sums, Parseval's identity, scalar product in terms of the Fourier coefficients. Almost everywhere convergence of the Fourier partial sums: Carleson's Theorem (only statement). Exercises 2.2 (Partial Fourier series in terms of trigonometric functions) and 2.3. | Pages 55-56 |
| Friday, March 27 | Examples: Fourier series of trigonometric polynomials, Fourier series of \(x\) and computation of the sum of inverse squares of the integers (Basel's problem), Fourier series of \(x^2\) (resp. \(e^x\)) and computation of the infinite sums \( \sum 1/n^4 \) (resp. \( \sum 1/(1+n^2) \)). Characterization of real valued functions via their Fourier coefficients. Convergence criteria for series in Hilbert spaces. | Pages 57-62 |
| Thursday, April 2 | Exercises 1.9 (\(L^\infty\) ball in \(L^2\) topology), 1.10 (extension of orthogonality condition from a dense subset), 1.11 ((un)boundedness of the identity map from \(L^1\) to \(L^2\)), 2.7 (Fouries series of the sign function), 2.8 (Fourier series of \(\pi-|x|\)). | Pages 46,59-60 and pages 193-196, 204-215 (solutions) |
| Friday, April 3-Friday, April 10 | No lectures, Easter holidays. | |
| Thursday, April 16-Friday, April 17 | Example 2.15 and Exercise 2.9 (Regularity of limit functions and convergence of derivatives), Exercise 2.11 (Fourier series of \(x^4-2x^2+1\)), Proposition 2.19 (Asymptotic behavior of Fourier coefficients of \(C^1\) functions), Corollary 2.20 (Absolute and uniform convergence of Fourier series), Theorem 2.22 (Fourier coefficients of higher derivatives, only statement), Corollary 2.25 (Uniform convergence of derivatives¸ only statement), Theorem 2.26 (Summability implies regularity). | Pages 62-69 |
| Thursday, April 23 | Dirichlet kernel: definition and properties (Proposition 2.30, Remark 2.31). Riemann-Lebesgue lemma (Lemma 2.32). | Pages 69-73 |
| Friday, April 24 | Pointwise convergence of Fourier series for Holder functions (Theorem 2.28). Recap and exercises on results regarding convergence of Fourier series and of the recostructed function (Section 2.5). Hurwitz's proof of the isoperimetric inequality (not examinable). | Pages 73-76 |
| Thursday, April 30 | Introduction to the heat equation and sketch of the strategy to find solutions using Fourier series. Derivation of the heat equation. Formal construction of a solution from the Fourier series. Statement of the existence of solutions' theorem (Theorem 2.34). | Pages 76-80 |
| Friday, May 1 | No lecture, Labour Day. | |
| Thursday, May 7-Friday, May 8 | Proof of the existence and the uniqueness of solution for the heat equation (Theorem 2.34, Theorem 2.37). Non-existence of solutions of the heat equation in the past (Example 2.39). Fourier Transform: definition and main properties (Definition 3.1, Theorem 3.3, Corollary 3.5) | Pages 80-86, 96-99 |
| Thursday, May 14 | No lecture, Ascension Day. | |
| Friday, May 15 | Basic properties of the Fourier Transform (Proposition 3.6), convolution product, convolution and Fourier Trasform (Proposition 3.8). Exercise and examples: Fourier transforms of radial functions (Exercise 3.1) and of the normal Gaussian distribution (Example 3.12, Example 3.16). Fourier transform of the derivative (Proposition 3.14, without proof) and derivative of the Fourier transform (Proposition 3.15, without proof). | Pages 99-106 |
| Thursday, May 21 | The space of Schwartz Functions and the Inversion Formula (Section 3.3). | Pages 106-111 |
| Friday, May 22 | The Fourier Transform in \(L^2\) (Theorem 3.31, Proposition 3.35), Plancherel's Identity (Theorem 3.32) and Heisenberg Inequality (Theorem 3.38). | Pages 112-117 |
| Thursday, May 28 | The heat equation and the Fourier transform (Section 3.6). | Pages 120-124 |
| Friday, May 29 | A compact overview on Spectral Theory. See the notes (protected with the same password of the lecture notes). |
A new exercise sheet will be posted here one week before each exercise class. The exercises will be related to topics covered in the previous week. The solutions will be posted by the end of the exercise class day.
Starting from Problem Set 2, each problem set includes a bonus exercise. Typically, it is an exam-level multiple choice exercise. If you hand in at least 9 out of 12 correctly solved bonus exercises, you will get extra points in the final mark. The bonus is worth 0.125 points. It can cumulate with the Analysis III bonus (also worth 0.125), for a total bonus up to 0.25. Deadline: to have the bonus exercise counted (and to submit the problem set), the deadline is Tuesday at midnight.
Things to keep in mind:
| Problem set | Due by | Upload link | Solutions |
|---|---|---|---|
| Problem set 1 | February 24 | Submission | Solutions problem set 1 |
| Problem set 2 | March 3 (23:59) | Submission | Solutions problem set 2 |
| Problem set 3 | March 10 (23:59) | Submission | Solutions problem set 3 |
| Problem set 4 | March 17 (23:59) | Submission | Solutions problem set 4 |
| Problem set 5 | March 24 (23:59) | Submission | Solutions problem set 5 |
| Problem set 6 | March 31 (23:59) | Submission | Solutions problem set 6 |
| Problem set 7 | April 14 (23:59) | Submission | Solutions problem set 7 |
| Problem set 8 | April 21 (23:59) | Submission | Solutions problem set 8 |
| Problem set 9 | April 28 (23:59) | Submission | Solutions problem set 9 |
| Problem set 10 | May 5 (23:59) | Submission | Solutions problem set 10 |
| Problem set 11 | May 12 (23:59) | Submission | Solutions problem set 11 |
| Problem set 12 | May 19 (23:59) | Submission | Solutions problem set 12 |
| Problem set 13 | May 26 (23:59) | Submission | Solutions problem set 13 |
| Problem set 14 | -- | -- | Solutions problem set 14 | Problem set 15 | -- | -- | Solutions problem set 15 |
The written exam, joint with Analysis III, will last complexively 3 hours and it will be a closed book exam -- no notes will be allowed. The oral exam, specific to Analysis IV, will last 20 minutes. The exam program is the same for the oral and written exam and is available here.
Here you can find a 90min mock exam that is resembles the Analysis IV part of the exam.
You can find the current version of the notes of the course here (password protected). Besides, below you find some textbooks that cover similar topics and that have been used to prepare lectures: