- Lecturer
- Mikaela Iacobelli
- Coordinator
- Marco Badran

This course will cover the following topics

- Real and complex Hilbert spaces, Hilbert bases and Riesz representation Theorem. See Chapter 4 of [AmDaMe] and Chapter I and III section 1-3 of [Co].
- Fourier series of a function in \(L^2([-\pi, \pi]; \mathbb{C})\), relationship between the regularity of a function and the asymptotic behaviour of the Fourier coefficients. Application to the resolution of linear partial differential equations with various boundary conditions in \([-\pi, \pi]\).
- Fourier Transform in \(\mathbb{R}^d\) and its elementary properties, relationship between the regularity of the function and the asymptotic behaviour of its Fourier transform, relationship between the summability of the function and the regularity of it Fourier transform. Application to the resolution of linear partial differential equations with various decay conditions in \(\mathbb{R}^d\).
- Compact operators on Hilbert spaces, Self-adjoint operators, the spectral theorem and applications to eigenvalue problems.

You can find all the relevant informations about the course in the course catalog.

Lectures take place in HG F3 on Wednesdays form 9:15 to 10:00 and on Fridays from 10:15 to 12:00. The first lecture will be on February 21st, 2024.

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 February 21st, 2024.

When | Where | Who | Language |
---|---|---|---|

Wednesday 10:15-12:00 | HG G 26.1 | D. Urech | German |

Wednesday 10:15-12:00 | LEE D 105 | C. Amend | English or German |

Wednesday 10:15-12:00 | ML F 40 | J. Conan | English |

Wednesday 10:15-12:00 | ML H 43 | U. Faure | English |

Wednesday 10:15-12:00 | ML J 34.1 | A. Künzi | English or German |

Wednesday 12:15-14:00 | ML F 40 | H. Töpel | 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 marco.badran@math.ethz.ch for an appointment.

The diary will be updated each week with the topics covered in class.

Date | Topic discussed | Notes |
---|---|---|

Wednesday, February 21 | Introduction to the course, definition of (real and complex) vector spaces and inner product spaces. Examples: \(\mathbb{R}^d\) , \(\mathbb{C}^d\), \(\ell^2(\mathbb{C})\). | |

Friday, February 23 | 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 1.24 (continuity of the inner product), exercise 1.2, example 1.27 (some norms on finite dimensional), example 1.28 (\(C([0,1])\) is an infinite dimensional vector space) and some classical norms. | |

Wednesday, February 28 | \(L^p\) 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, definition of Hilbert spaces, examples of Hilbert spaces (\(\mathbb{C}^d\), \(\ell^2\), \(L^2\)) | |

Friday, March 1 | 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, example of a non separable Hilbert space, basis of a Hilbert space, definition of orthonormal set for a Hilbert space. Statement of Bessel inequality and Parseval identity. | |

Wednesday, March 6 | Proof of the theorem about orthonormal sets of a Hilbert space (Bessel inequality, Parseval identity), Hilbert basis, Completeness criterion for separable Hilbert Spaces. | |

Friday, March 8 | Theorem of existence of an Hilbert basis, example of \(\ell^2\), and example of compactly supported sequences. Every separable Hilbert space of infinite dimension is isometric to \(\ell^2\), Hilbert bases are not algebraic bases, theorem about the existence of the projection on closed subspaces and characterisation of the orthogonal projection. Statement and proof of the theorem about the projection on closed convex set. Characterization of the projection onto a convex set, with proof. | |

Wednesday, March 13 | Nontriviality of the complement of a proper closed subspace. Projection onto a subspace in terms of a Hilbert basis of the subspace. Every closed subspace has an orthogonal complement. Proposition about isometry of \(H\) and \(Y\oplus Y^\perp\) for any closed subspace \(Y\). Definition of linear and bounded operators, example of an unbounded operator (derivative). | |

Friday, March 15 | 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, with proof. Corollary: canonical isometric isomorphism between a Hilbert space and its dual. Application: Von Neumann’s proof of the Radon-Nikodym theorem (not examinable). | |

Wednesday, 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). | |

Friday, March 22 | 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. Example: Fourier series of sines and cosines. Proposition: characterization of real valued functions via their Fourier coefficients. Almost everywhere convergence of the Fourier partial sums, Carleson's Theorem (only statement). Examples: Foureir series of trigonometric polynomial, Fourier series of \(x\) and computation of the sum of inverse squares of the integers (Basel's problem) | |

Wednesday, March 27 | Convergence criteria for series in Hilbert spaces. Example: regularity of limit functions. Fourier coefficients of the derivative. Asymptotic behaviour of Fourier coefficients of \(C^1\) functions. | |

Wednesday, April 10 | Corollary about uniform convergence of the Fourier series of a \(C^1\) function. Fourier coefficients of higher derivatives of a function. Corollary: uniform convergence of the Fourier series of a function \(C^h\) with all its derivatives. Theorem: summability implies regularity (statement only). | |

Friday, April 12 | Proof of the summability implies regularity theorem. Statement of the pointwise convergence theorem. Definition of Dirichlet kernel and properties (mean value and explicit expression). Riemann-Lebesgue lemma. Proof of pointwise convergence theorem. Introduction to the heat equation and sketch of the strategy to find solutions using Fourier series. | |

Wednesday, April 17 | Derivation of the heat equation. Formal construction of a solution from the Fourier series. Statement of the existence of solutions' theorem. | |

Friday, April 19 | Existence of a solution to the heat equation if the initial datum has summable Fourier coefficients. Uniqueness of solutions to the heat equation. Example: non-existence in the past. Definition of the Fourier transform in \(\mathbb{R}^d\). Properties of the Fourier transform (only statement). Corollary: \(\mathcal{F}\colon L^1(\mathbb{R}^d)\to \mathcal{C}_0(\mathbb{R}^d)\) is a bounded linear operator. | |

Wednesday, April 24 | Properties of Fourier transform (Theorem 3.3, Proposition 3.6). Convolution and Fourier transform. | |

Friday, April 26 | Examples: computation of the Fourier transform of a few functions ( \(\exp(-|x|)\), Gaussian in \(d\) variables). Theorems: the Fourier transform transforms directional derivatives in multiplications by the respective coordinate function, and vice versa. Application: alternative computation of the Fourier transform of the Gaussian via ODE. | |

Friday, May 3 | Definition of the Schwartz class of functions. Theorem: the Fourier transform of a Schwartz function is itself a Schwartz function, and the formulas realting directional derivatives and multiplication by coordinate functions holds. Theorem: the inversion formula for Schwartz functions (with proof). Generalisations to functions such that both \(f\) and \(\hat f\) are in \(L^1(\mathbb{R}^d)\). | |

Wednesday, May 8 | The Fourier Transform is an Isometry on \(L^2\), Plancherel’s Identity. | |

Friday, May 10 | Formula for the Fourier transform of the derivative of a \(L^2\) function, The Heisenberg inequality, derivation of the 1d Heat equation from a random walk (non examinable), the heat equation heuristic and definition of Heat kernel. | |

Wednesday, May 15 | The heat equation via Fourier transform. | |

Friday, May 17 | Compact operators. Characterization of compact operators (without proof). Finite rank operators. The space of compact operators is a closed subspace of the space of linear operators. Integral operators. Adjoint operator. Existence and uniqueness of the adjoint operator. An operator is compact if and only if its adjoint is compact (without proof). | |

Wednesday, May 22 | (no proofs) Eigenvalue and eigenvectors. Fredholm alternative I, II, III and IV. | |

Friday, May 24 | (no proofs) Resolvent and spectrum. Open mapping theorem and corollaries. Banach fixed point theorem. Structure theorem of the spectrum of compact operators. | |

Wednesday, May 29 | Exercise class. | |

Friday, May 31 | (no proofs) The spectral theorem and applications. See Federico Franceschini's 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 1 you will find a **bonus** exercise in every Problem Set. Typically, it will be an exam-level multiple choice exercise. Those who will hand in **at least 9 out of 11** correctly solved bonus exercise will get extra points in the final mark.
The bonus is worth 0.125 points, to potentially cumulate with the Analysis III bonus, also worth 0.125, for an extra 0.25 in the final mark. The deadline for submitting the solutions to Problem Set and having the bonus exercise counted is Tuesday at midnight.

Things to keep in mind:

- With the exeption of the bonus exercises, handing in the solutions gives no points for the final exam, although working on the problem sets will greatly increase your chances to pass the exam.
- Problems are
*loosely*ordered by increasing difficulty. You do not have to do all of them, some can be rather challenging. Work in groups is encouraged. - The purpose of giving you problem sets is to help you learn the material and develop your problem solving skills.
- The purpose of giving you the solutions and discuss them in exercise classes is to show you how to solve the problems you could not solve. Or to show you more efficient proofs.
- The purpose of letting you hand in the solutions and us correcting them
*is not*assess your problem solving skills, but give you a feedback on how you write mathematics.

Problem set | Due by | Upload link | Solutions |
---|---|---|---|

Problem set 0 | February 28 | Submission | Solutions |

Problem set 1 | March 6 | Submission | Solutions |

Problem set 2 | March 12 | Submission | Solutions |

Problem set 3 | March 19 | Submission | Solutions |

Problem set 4 | March 26 | Submission | Solutions |

Problem set 5 | April 9 | Submission | Solutions |

Problem set 6 | April 16 | Submission | Solutions |

Problem set 7 | April 23 | Submission | Solutions |

Problem set 8 | May 7 | Submission | Solutions |

Problem set 9 | May 14 | Submission | Solutions |

Problem set 10 | May 21 | Submission | Solutions |

Problem set 11 | May 28 | Submission | Solutions |

Problem set 12 | June 4 | Submission | Solutions |

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. Here you will find the solutions to the mock exam.

You can find the lastest version of the course script here (password protected). Besides, below you find some textbooks that cover similar topics and that have been used to prepare lectures:

- [AmDaMe] "Introduction to measure Theory and Integration" by Ambrosio, Da Prato and Mennucci (Edizioni della Normale, 2011).
- [Bo] "Méthodes mathématiques pour les sciences physiques" by Bony (École polytechnique, 2000).
- [CaDA] "Introduction to Measure Theory and Functional Analysis" by Cannarsa and D'Aprile (Springer, 2008).
- [Co] "A Course in Functional Analysis" by Conway (Springer, 2007).
- [Ev] "Partial Differential Equations" by Evans (American Mathematical Society, 2010).
- [Kr] "Introductory Functional Analysis with Applications" by Kreyszig (John Wiley & Sons, 1978).
- [StSh] "Fourier Analysis: an Introduction" by Stein and Shakarchi (Princeton University Press, 2003)
- [Ci] "Exercises and Problems in Mathematical Methods of Physics" by Cicogna (Springer, 2020)