- Lecturer
- Prof. Dr. Alessandro Carlotto
- Course Assistant
- Giada Franz
- Teaching Assistants
- Riccardo Caniato, Federico Franceschini, Filippo Gaia, Salome Schumacher

- Lectures
- Mon 10-12 / HG G 5 - Livestream

Thu 14-16 / HG G 5 - Livestream - Exercise classes
- Mon 9-10
- Office hours
- Thu 16-18
- First lecture
- 17.09.2020
- Course Catalogue
- 401-3461-00L Functional Analysis I

Starting from Monday, November 2nd, all our teaching activities are transferred online.

Solid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with topology and measure theory, in part. Lebesgue integration and \( L^p \) spaces).

Baire category; Banach and Hilbert spaces, bounded linear operators; basic principles: Uniform boundedness, open mapping/closed graph theorem, Hahn-Banach; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces.

Michael Struwe. *Funktionalanalysis I und II*. Lecture notes, ETH Zürich, 2019/20.

Haim Brezis. *Functional analysis, Sobolev spaces and partial differential equations*. Universitext. Springer, New York, 2011.

Michael Reed and Barry Simon. *Methods of Modern Mathematical Physics – Volume 1 (Functional Analysis).* Academic Press, 1981

Elias M. Stein and Rami Shakarchi. *Functional analysis (volume 4 of Princeton Lectures in Analysis).* Princeton University Press, Princeton, NJ, 2011.

Peter D. Lax. *Functional analysis.* Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 2002.

Walter Rudin. *Functional analysis.* International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.

Terence Tao. *Ask yourself dumb questions – and answer them!*

Terence Tao. *Think ahead*.

Paul R. Halmos. *How to write Mathematics*.

Susan Hermiller, Melanie Martin and Eric York. *Learning Calculus*.

The live streaming of the lecture is available at the link https://video.ethz.ch/live/lectures/zentrum/hg/hg-g-5.html. After each lecture the recording is published at https://video.ethz.ch/lectures/d-math/2020/autumn/401-3461-00L.html.

For further information, visit https://ethz.ch/services/en/it-services/catalogue/multimedia/lecture-recording/stud.html.

Date | Content | Notes | Reference (Struwe) | Extras | |
---|---|---|---|---|---|

1 | 17.09. | General presentation of the course and its scopes. Textbooks and other resources. | Notes - L01 | What is Functional Analysis? | |

2 | 21.09. | Review on metric spaces and completeness. The Baire lemma, contextualisation and proof. Pointwise versus uniform boundedness of maps defined on metric spaces. | Notes - L02 | Sections 1.1, 1.2, 1.3 (up to Def. 1.3.1) and Satz 1.4.3 | |

3 | 24.09. | The Banach-Steinhaus theorem (i.e. uniform boundedness principle for linear maps). Non-existence of countable algebraic bases of infinite-dimensional Banach spaces. Baire category, some simple facts. Different notions of smallness for subsets of the real line. | Notes - L03 | Satz 1.4.3 and Section 1.3 | Any algebraic basis is uncountable, Cantor set (with video) |

4 | 28.09. | The (dis-)continuity set of a function. Application: there is no function that is "only" discontinuous at irrational points. Normed and Banach spaces, examples. Equivalent and inequivalent norms. | Notes - L04 | Sections 1.4 and 2.1 (up to. Satz 2.1.3) | |

5 | 01.10. | Basic facts about subspaces, closure and complements. The Riesz projection lemma, non-compactness of the unit sphere in Banach spaces. Equivalent continuity criteria for linear maps, examples. | Notes - L05 | Sections 2.1, 2.2 (up to Satz 2.2.3) | Equivalent notions of compactness |

6 | 05.10. | The Banach algebra of bounded linear operators. Series of bounded linear operators. Two examples: the exponential of an operator, building an inverse through the Neumann series. Product of Banach spaces: definitions and a collection of basic facts. | Notes - L06 | Sections 2.2 (up to Satz 2.2.6), 2.5 | |

7 | 08.10. | Spectral radius. The linear group of bijective bounded linear operators. The quotient of a Banach space by a closed subspace. | Notes - L07 | Sections 2.2, 2.3 | |

8 | 12.10. | Hilbert spaces, examples and basic facts. Orthogonal subspaces. Existence of orthogonal projections, equivalent characterization (least distance property). | Notes - L08 | Section 2.4 | |

9 | 15.10. | Orthonormal systems, Bessel's inequality, Fourier series, Parseval identity. Hilbertean bases, the separability criterion, isometric classification. | Notes - L09 | Hilbertian bases | |

10 | 19.10. | The Banach-Steinhaus theorem, a counterexample with an incomplete domain. The open mapping principle, linear homeomorphisms between Banach spaces. | Notes - L10 | Sections 3.1, 3.2 | |

11 | 23.10. | The closed graph theorem. The Töplitz criterion for symmetric maps. The case of unbounded operators, the continuous inverse theorem. | Notes - L11 | Section 3.3 | |

12 | 26.10. | Closable operators. Bounded operators are closable. Example of a non-closable operator. Smooth differential operators are closable. | Notes - L12 | Section 3.4 | |

13 | 29.10. | The Hahn-Banach theorem over the real and the complex field. Extension of linear, continuous functionals. | Notes - L13 | Section 4.1 | |

14 | 02.11. | The dual of a Banach space, basic facts. Geometric versions of Hahn-Banach. Annihilator of a subspace. | Notes - L14 | Section 4.2 | |

15 | 05.11. | Duality in Hilbert spaces: the Riesz representation theorem. Bilinear forms and the Lax-Milgram theorem. | Notes - L15 | Section 4.3 | |

16 | 09.11. | Duality for \(L^p\) spaces when \(p\) is finite, discussion about \(L^\infty\). Uniform convexity of the unit ball: the Clarkson inequalities. | Notes - L16 | Section 4.4 | |

17 | 12.11. | The Minkowski functional of an open, convex set. Separation of convex sets. Extremal points, an existence result and the Krein-Milman theorem. | Notes - L17 | Section 4.5 | |

18 | 16.11. | Weak convergence: motivation, basic facts and examples. The weak topology on a normed space, different notions of closure. | Notes - L18 | Section 4.6 | |

19 | 19.11. | The bidual of a normed space. The canonical embedding of a normed space into its bidual. Reflexivity: definitions, examples, criteria. A closed subspace of a reflexive space is reflexive. | Notes - L19 | Section 5.1 | |

20 | 23.11. | Separability: examples and basic criteria. The weak*-topology on the dual of a normed space. The Banach-Alaoglu theorem. | Notes - L20 | Sections 5.2 and 5.3 (up to Satz 5.3.2) | |

21 | 26.11. | The Eberlein-Smulyan compactness theorem. Minimizing distance from a point to a convex set. Lower semicontinuity, coercivity and the direct method of the Calculus of Variations, examples. | Notes - L21 | Sections 5.3 and 5.4 | |

22 | 30.11. | The dual (adjoint) of a linear operator. The adjunction is an isometry on bounded operators. Orthogonality relations and solvability criteria for linear equations in Banach spaces. | Notes - L22 | Sections 6.1 and 6.2 (up to Def. 6.2.1) | |

23 | 03.12. | Compact operators, basic properties. Operators of the form \(id−T\) with \(T\) compact. Two strong compactness theorems: Arzelà-Ascoli and Fréchet-Kolmogorov. Related examples of compact operators. | Notes - L23 | Sections 6.2 and 6.3 | |

24 | 07.12. | Dual vs. adjoint operator in Hilbert spaces. Symmetric and self-adjoint operators, examples. Fredholm operators. Resolvent and spectrum of a linear map. The spectrum is closed. Properties of the resolvent. | Notes - L24 | Sections 6.4 and 6.5 (up to Def. 6.5.3) | |

25 | 10.12. | Partition of the spectrum: point, continuous, residual. Three examples. Characterization of the spectral radius for bounded operators. Functional calculus for rational functions. | Notes - L25 | Section 6.5 | |

26 | 14.12. | Eigenvalues of symmetric operators are real, but their spectrum may equal \(\mathbb C\). Self-adjoint operators have real spectrum and related results. Normal operators, characterization of their spectral radius. | Notes - L26 | Section 6.6 | |

27 | 17.12. | Normal operators, characterization of their spectral radius. The spectral theorem for compact, self-adjoint operators. Courant-Fischer characterization of eigenvalues. | Notes - L27 | Section 6.7 | Spectral theory over the real field |

- Date
- 13.02.2021
- Time
- 9:00-12:00
- Place
- HG F 5 (for students having identification numbers from 001 to 025)

HG F 7 (for students having identification numbers from 026 to 070)

- Rules of the exam
- Advice for the exam
- Probeprüfung -> solutions of the Probeprüfung
- Training problems:

Final exam 2017/2018 -> solutions of the Final exam 2017/2018

Repetition exam 2017/2018 -> solutions of the Repetition exam 2017/2018

In order to easily interact, we set up a forum for our course at the link Functional Analysis I (Autumn 2020) - Forum. You have to sign up with your ETH credentials. There you find several topics where you can ask questions and discuss about the lectures, the problem sets, the exam, etc. Use it!

Please register and enroll for a teaching assistant in myStudies. The enrollment is needed to attend the exercise class and to hand in your homework.

All the exercise classes have now migrated online. Below you find the link to the Zoom meetings of each exercise class (accessible with the password we sent you by email).

Assistant | Room |
---|---|

Riccardo Caniato | ETH Zoom |

Federico Franceschini | ETH Zoom |

Filippo Gaia | ETH Zoom |

Salome Schumacher | ETH Zoom |

Here is the diary of the exercise classes. Salome's online exercise classes are recorded, the videos are accessible at the links below (with the password that we sent you by email) and the notes are available in polybox - Functional Analysis I (with the same password).

Date | Content | Recordings | |
---|---|---|---|

1 | 21.09. | Review on scalar product, norm, distance, topology and the relations among these notions. Discussion of problems 1.1 and 1.5. | Video - EC01 |

2 | 28.09. | Discussion of problems 1.2 (i), 1.3 and 2.4. Nonexistence of function continuous only at rational numbers. | Video - EC02, Video - Ex 2.4 |

3 | 05.10. | Discussion of problems 2.5 and 3.3. | Video - EC03 |

4 | 12.10. | Topological complements in a Banach space: when a subspace is complementable. Discussion of problems 3.4 and 4.2. | Video - EC04 |

5 | 19.10. | Discussion problems 4.3 and 5.2. | Video - EC05 |

6 | 26.10. | Discussion of problems 5.3, 5.4 and 5.6. | Video - EC06 |

7 | 02.11. | Discussion of problems 6.2, 6.3 and 7.2. | Video - EC07 |

8 | 09.11. | Discussion of problems 7.5 and 8.1. | Video - EC08 |

9 | 16.11. | Discussion of problems 7.6, 7.4 and 9.1. | Video - EC09 |

10 | 23.11. | Discussion of problems 9.5, 9.3 and 10.2. | Video - EC10 |

11 | 30.11. | Discussion of problems 10.4, 11.3 and 11.1. | Video - EC11 |

12 | 07.12. | Discussion of problems 11.4, 11.5 and 12.4. | Video - EC12 |

13 | 14.12. | Discussion of problem 12.5, the Courant-Fischer characterization of eigenvalues. | Video - EC13 |

Every Friday, a new problem set is uploaded here. You have ten days to solve the problems and hand in your solutions via the platform SAMUpTool, for grading. During exercise classes on Monday some of the problems are discussed.

Every problem is marked by one of the following symbols.

- Computation
- Get your hands dirty and calculate.
- Bookkeeping
- Apply what you learn in basic situations.
- Comprehension
- Construct examples and give full proofs.
- Previous exam
- Exercise given in an old exam.

- Hard problem
- Challenging problems are denoted by one up to three diamonds. It is recommended that you start working on these problems only after you have reviewed the weekly material and carefully solved all other exercises in the assignment.

Assignment date | Due date | Problem set | Solution |
---|---|---|---|

Thu 17.09. | Mon 28.09. | Problem set 1 | Solutions 1 |

Mon 28.09. | Mon 05.10. | Problem set 2 | Solutions 2 |

Mon 05.10. | Mon 12.10. | Problem set 3 | Solutions 3 |

Mon 12.10. | Mon 19.10. | Problem set 4 | Solutions 4 |

Mon 19.10. | Mon 26.10. | Problem set 5 | Solutions 5 |

Fri 23.10. | Mon 02.11. | Problem set 6 | Solutions 6 |

Fri 30.10. | Mon 09.11. | Problem set 7 | Solutions 7 |

Fri 06.11. | Mon 16.11. | Problem set 8 - Hints | Solutions 8 |

Fri 13.11. | Mon 23.11. | Problem set 9 - Hints | Solutions 9 |

Fri 20.11. | Mon 30.11. | Problem set 10 - Hints | Solutions 10 |

Fri 27.11. | Mon 07.12. | Problem set 11 - Hints | Solutions 11 |

Fri 04.12. | Mon 14.12. | Problem set 12 - Hints | Solutions 12 |

Fri 11.12. | Mon 21.12. | Problem set 13 - Hints | Solutions 13 |

You are free to come and ask questions. The office hours are held via Zoom starting from October 29th. The schedule is as follows (up to possible short-term changes, please check for updates).

Date | Time | Location | Assistant |
---|---|---|---|

Thu 24.09. | 16-18 | HG G 28 | Giada Franz |

Thu 01.10. | 16-18 | HG F 28.3 | Riccardo Caniato |

Thu 08.10. | 16-18 | HG F 28.3 | Federico Franceschini |

Thu 15.10. | 16-18 | HG F 27.9 | Filippo Gaia |

Thu 22.10. | 16-18 | HG F 28.1 | Salome Schumacher |

Thu 29.10. | 16-18 | ETH Zoom | Giada Franz |

Thu 05.11. | 16-18 | ETH Zoom | Riccardo Caniato |

Thu 12.11. | 16-18 | ETH Zoom | Federico Franceschini |

Thu 19.11. | 16-18 | ETH Zoom | Filippo Gaia |

Thu 26.11. | 16-18 | ETH Zoom | Salome Schumacher |

Thu 03.12. | 16-18 | ETH Zoom | Giada Franz |

Thu 10.12. | 16-18 | ETH Zoom | Riccardo Caniato |

Thu 17.12. | 16-18 | ETH Zoom | Federico Franceschini |

Mon 01.02. | 14-15 | ETH Zoom | Salome Schumacher |

Mon 08.02. | 14-15 | ETH Zoom | Federico Franceschini |