- Lecturer
- Prof. Dr. Alessandro Carlotto
- Course Assistant
- Mario Schulz
- Teaching Assistants
- Yash Jhaveri
- Kathrin Näf
- Alessandro Pigati
- Shengquan Xiang

- Lectures
- Mon 13–15 / HG G 3
- Wed 08–10 / HG G 5
- Exercise classes
- Mon 09–10
- Office hours
- Fr 13–15 or 14–16
- Thu 14–16 or 16–18

- First Lecture
- 20.09.2017
- First Exercise class
- 25.09.2017
- Course Catalogue
- 401-3461-00L Functional Analysis I

Michael Struwe. *Funktionalanalysis I und II*. Lecture notes, ETH Zürich, 2013/14.

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*.

Date | Content | Reference 1 (Struwe) | Reference 2 (Brezis) | Extras |
---|---|---|---|---|

20.09.2017 | General presentation of the course and its scopes. Textbooks and other resources. |
– | – | What is Functional Analysis? |

25.09.2017 | The Baire Lemma, contextualization and proof. Two questions in Real Analysis. |
§ 1.1, § 1.2, § 1.3 (up to Def. 1.3.1.) |
§ 2.1 | – |

27.09.2017 | Uniform boundedness principle in metric spaces. Baire category, different notions of smallness. |
Satz 1.4.3 and § 1.3 | § 2.2 | – |

02.10.2017 | Normed and Banach spaces, examples, equivalent and inequivalent norms. Closed subspaces, non-compactness of the unit sphere in a Banach space. |
§ 2.1 | – | – |

04.10.2017 | Continuity criteria for linear maps. The Banach algebra of continuous, linear maps. |
§ 2.2 (up to Satz 2.2.5) |
– | – |

09.10.2017 | Series of linear, bounded operators. Spectral radius. Invertibility. Product of Banach spaces. Intersections of Banach subspaces. |
§ 2.2, § 2.5 | – | – |

11.10.2017 | Quotient of a Banach space by a closed subspace. Hilbert spaces, orthogonal complements, projections. |
§ 2.3, § 2.4 | § 5.1 | – |

16.10.2017 | Orthonormal systems, Bessel's inequality, abstract Fourier series. Hilbertean bases, the separability criterion, isometric classification. |
– | § 5.4 | Hilbertean bases and applications |

18.10.2017 | The Banach-Steinhaus theorem, a counterexample with an incomplete domain. The open mapping principle, linear homeomorphisms between Banach spaces. |
§ 3.1, § 3.2 | § 2.2, § 2.3 | – |

23.10.2017 | The closed graph theorem. The Töplitz criterion for symmetric maps. The case of unbounded operators, the continuous inverse theorem. |
§ 3.3 | § 2.3 | – |

25.10.2017 | Closable operators. Bounded operators are closable. Example of a non-closable operator. Smooth differential operators are closable. |
§ 3.4 | – | – |

30.10.2017 | The Hahn-Banach theorem over the real and the complex fields. Extension of linear, continuous functionals. |
§ 4.1 | § 1.1 | – |

01.11.2017 | The dual of a Banach space, basic facts. Geometric versions of Hahn-Banach. Annihilator of a subspace. |
§ 4.2 | § 1.2, § 2.5 | – |

06.11.2017 | Duality in Hilbert spaces: the Riesz representation theorem. Bilinear forms and the Lax-Milgram theorem. |
§ 4.3 | § 5.2, § 5.3 | – |

08.11.2017 | Duality for \(L^p\) spaces when \(p\) is finite, discussion about \(L^\infty\). Uniform convexity of the unit ball: the Clarkson inequalities. |
§ 4.4 | § 4.3 | – |

13.11.2017 | The Minkowski functional of an open, convex set. Separation of convex sets. Extremal points, an existence result and the Krein-Milan theorem. |
§ 4.5 | § 1.2 | – |

15.11.2017 | Weak convergence, basic facts and examples. The weak topology on a normed space, different notions of closure. |
§ 4.6 | § 3.1, § 3.2 | – |

20.11.2017 | 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. |
§ 5.1 | § 3.5 | – |

22.11.2017 | Separability: examples and basic criteria. The weak*-topology on the dual of a normed space. The Banach-Alaoglu theorem. |
§ 5.2, § 5.3 (up to Satz 5.3.2) |
§ 3.6, § 3.4 | – |

27.11.2017 | 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. |
§ 5.3, § 5.4 | § 3.5, § 3.6 | – |

29.11.2017 | 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. |
§ 6.1, § 6.2 (up to Def. 6.2.1) |
§ 2.6, § 2.7 | – |

04.12.2017 | Compact operators, basic properties. Operators of the form \(\mathrm{id}-T\) with \(T\) compact. Two strong compactness theorems: Arzelà-Ascoli and Fréchet-Kolmogorov. |
§ 6.2, § 6.3 | § 6.1, § 6.2 | – |

06.12.2017 | Dual vs. adjoint operator in Hilbert spaces. Symmetric and self-adjoint operators, examples. Resolvent and spectrum of a linear map. The spectrum is closed. Properties of the resolvent. |
§ 6.4, § 6.5 (up to Def. 6.5.3) |
§ 6.3 | – |

11.12.2017 | Partition of the spectrum: point, continuous, residual. Three examples.
Characterization of the spectral radius for bounded operators. Functional calculus for rational functions. |
§ 6.5 | – | – |

13.12.2017 | Eigenvalues of symmetric operators are real, but their spectrum may equal \(\mathbb{C}\). Self-adjoint operators have real spectrum and related results. |
§ 6.6 | – | – |

18.12.2017 | Normal operators, characterization of their spectral radius.
The spectral theorem for compact, self-adjoint operators. Courant-Fischer characterization of eigenvalues. |
§ 6.7 | § 6.3, § 6.4 | – |

20.12.2017 | Spectral theory over the real field: two perspectives and their equivalence. The Dirichlet spectrum of the Laplacian: some results and open problems. |
– | – | Complex to real spectrum |

Rules of the exam: | Rules.pdf |

Advice for the exam: | Advice.pdf |

Probeprüfung: | Probeprüfung and Solutions of the Probeprüfung |

Training problems: | Training.pdf |

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

Thursday | 11.01.2018 | 14:00–16:00 | HG G 19.2 | Mario Schulz |

Thursday | 18.01.2018 | 14:00–16:00 | HG G 19.2 | Alessandro Pigati |

Monday | 22.01.2018 | 14:00–16:00 | HG G 19.2 | Alessandro Pigati,
Mario Schulz |

Please register in myStudies and use echo.ethz.ch to enroll for an exercise class.

Assistant | Classroom |
---|---|

Yash Jhaveri | HG G 26.1 |

Kathrin Näf | HG F 26.3 |

Alessandro Pigati | HG F 26.5 |

Shengquan Xiang | HG E 21 |

Every Monday, a new problem set is uploaded here. You have one week time to solve the problems. The following Monday during exercise class the corresponding solutions are discussed and you are invited to hand in your work for grading. If you have difficulties understanding or solving certain tasks, please prepare your questions and join the office hours on Thursdays or Fridays.

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.

- 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 Sets | Extra Hints | Solutions | Comments |
---|---|---|---|---|---|

20.09.2017 | 25.09.2017 | Problem Set 1 | – | Solution 1 | – |

25.09.2017 | 02.10.2017 | Problem Set 2 | – | Solution 2 | – |

02.10.2017 | 09.10.2017 | Problem Set 3 | – | Solution 3 | – |

09.10.2017 | 16.10.2017 | Problem Set 4 | – | Solution 4 | – |

16.10.2017 | 23.10.2017 | Problem Set 5 | – | Solution 5 | 5.1 (b): \(A\) and \(B\) must be symmetric. |

23.10.2017 | 30.10.2017 | Problem Set 6 | – | Solution 6 | The tools needed to solve problems 6.3 – 6.7 will be covered on Wednesday, 25.10.2017. |

30.10.2017 | 06.11.2017 | Problem Set 7 | Hints 7 | Solution 7 | – |

06.11.2017 | 13.11.2017 | Problem Set 8 | Hints 8 | Solution 8 | Uniform convexity |

13.11.2017 | 20.11.2017 | Problem Set 9 | Hints 9 | Solution 9 | – |

20.11.2017 | 27.11.2017 | Problem Set 10 | Hints 10 | Solution 10 | – |

27.11.2017 | 04.12.2017 | Problem Set 11 | Hints 11 | Solution 11 | – |

04.12.2017 | 11.12.2017 | Problem Set 12 | Hints 12 | Solution 12 | – |

11.12.2017 | 18.12.2017 | Problem Set 13 | Hints 13 | Solution 13 | – |

**UPDATE:** From November on, office hours are on Thursdays and **Fridays**.

You are free to come and ask questions. The location changes depending on which assistant is on duty. The schedule is as follows (up to possible short-term changes, please check for updates):

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

Thursday | 21.09.2017 | 14:00–16:00 | HG F 28.3 | Mario Schulz |

Tuesday | 26.09.2017 | 13:00–15:00 | HG FO 28.7 | Kathrin Näf |

Thursday | 28.09.2017 | 16:00–18:00 | HG FO 27.9 | Alessandro Pigati |

Tuesday | 03.10.2017 | 13:00–15:00 | HG G 38.1 | Shengquan Xiang |

Thursday | 05.10.2017 | 14:00–16:00 | HG G 28 | Yash Jhaveri |

Tuesday | 10.10.2017 | 13:00–15:00 | HG FO 28.7 | Kathrin Näf |

Thursday | 12.10.2017 | 16:00–18:00 | HG FO 27.9 | Alessandro Pigati |

Tuesday | 17.10.2017 | 13:00–15:00 | HG G 38.1 | Shengquan Xiang |

Thursday | 19.10.2017 | 14:00–16:00 | HG G 28 | Yash Jhaveri |

Tuesday | 24.10.2017 | 13:00–15:00 | HG FO 28.7 | Kathrin Näf |

Thursday | 26.10.2017 | 16:00–18:00 | HG FO 27.9 | Alessandro Pigati |

Thursday | 02.11.2017 | 14:00–16:00 | HG G 28 | Yash Jhaveri |

Friday | 03.11.2017 | 14:00–16:00 | HG G 38.1 | Shengquan Xiang |

Thursday | 09.11.2017 | 16:00–18:00 | HG FO 27.9 | Alessandro Pigati |

Friday | 10.11.2017 | 13:00–15:00 | HG F 28.3 | Mario Schulz |

Thursday | 16.11.2017 | 14:00–16:00 | HG G 28 | Yash Jhaveri |

Friday | 17.11.2017 | 14:00–16:00 | HG G 38.1 | Shengquan Xiang |

Thursday | 23.11.2017 | 16:00–18:00 | HG FO 27.9 | Alessandro Pigati |

Friday | 24.11.2017 | 13:00–15:00 | HG FO 28.7 | Kathrin Näf |

Thursday | 30.11.2017 | 14:00–16:00 | HG G 28 | Yash Jhaveri |

Friday | 01.12.2017 | 14:00–16:00 | HG G 38.1 | Shengquan Xiang |

Thursday | 07.12.2017 | 16:00–18:00 | HG FO 27.9 | Alessandro Pigati |

Friday | 08.12.2017 | 13:00–15:00 | HG FO 28.7 | Kathrin Näf |

Thursday | 14.12.2017 | 14:00–16:00 | HG G 28 | Yash Jhaveri |

Friday | 15.11.2017 | 14:00–16:00 | HG G 38.1 | Shengquan Xiang |

Thursday | 21.12.2017 | 14:00–16:00 (update) |
HG F 28.3 | Mario Schulz |

Friday | 22.12.2017 | 13:00–15:00 | HG FO 28.7 | Kathrin Näf |