WS 20/21: ** Functional Analysis ** (tutorials)

** This course has come to the end. See student evaluation reports (in Polish). **

**Lecutre**: prof. Piotr Rybka (Thursdays, 10:15 - 11:45)

**Tutorial**: Kuba Skrzeczkowski (Thursdays, 12:15 - 13:45)

**USOS**: link

This is a basic course for 3rd year students introducting fundamental concepts of normed spaces, Banach spaces and Hilbert spaces, providing general framework to study infinite dimensional spaces as well as operators acting on them that one meets in modern theory of Probability, PDEs, Numerical Analysis or Optimization.

** Grading**: at most 30 points collected from any of these:

- (H) homeworks - 12 series of 2 problems each week
**submitted in groups of 2**= max 24 points, - (A) active class participation - 13 classes x 1 point per class = max 13 points (this includes turning on your audio, video - 0.5 AND asking questions, answers etc - 0.5),
- (S) special problems - around 8 problems = max 32 points (each problem worth 4 points, be ready for a discussion about your solution!).

** Admission to the final exam ** after collecting either

- 12 points from homeworks (H) and 7 points from active class participation (A) or
- 20 points from special problems (S).

** Textbook**: There is an excellent textbook by Haim Brezis that contains the relevant theory, intuitive explanations and problems to solve (both easy and hard).
Some people recommend the book by Rudin but I find it too extensive, too difficult to understand and too general for the first course in Functional Analysis (especially if you are more applications-oriented).

** Last year tutorial **:
Here is a link to the last year tutorial. There are some materials (problems, solutions) that may be helpful. Problems discussed this year will be similar but not exactly the same (so far, I skipped Phillips Lemma and invertibility conditions).

** COVID related info ** : We gonna meet on Zoom each Thursday. You are expected to turn on your camera and audio (in case you don't have appropriate equipment contact me before 15/10).
I will upload content of the iPad blackboard and videos from the class below. Each week I will offer online office hours on Zoom. To compensate for the social distancing, you are expected
to submit homeworks in groups of 2. You should discuss content of the lecture and the tutorial with your friend.

** Problem Sets ** :

- PS1: normed spaces, Banach spaces, examples
- PS2: operators and their norms
- PS3: Banach-Steinhaus Theorem
- PS4: Closed Graph Theorem and Inverse Mapping Theorem
- PS5: Introduction to Hilbert spaces
- PS6: Dual spaces and Hahn-Banach theorems
- PS7: Introduction to weak convergence
- PS8: Orthonormal sets and basis in Hilbert spaces
- PS9: Spectrum of operators on Hilbert spaces
- PS10: Adjoint and self-adjoint operators on Hilbert spaces
- PS11: Compact operators, spectral theory
- PS12: Convolutions, density of smooth functions and Schwartz spaces
- PS13: Fourier transform

** My notes (maybe helpful?) ** :

- PS1: normed spaces, Banach spaces, examples (ver: 5.11.2020)
- PS2: operators and their norms (ver: 5.11.2020)
- PS3: Banach-Steinhaus Theorem (ver: 26.11.2020)
- PS4: Closed Graph Theorem and Inverse Mapping Theorem (ver: 26.11.2020)
- PS5: Introduction to Hilbert spaces (ver: 26.11.2020)
- PS6: Dual spaces and Hahn-Banach theorems (ver: 17.12.2020)
- PS7: Introduction to weak convergence (ver: 10.12.2020)
- PS8: Orthonormal sets and basis in Hilbert spaces (ver: 17.12.2020)
- PS9: Spectrum of operators on Hilbert spacess (ver: 14.01.2021)
- PS10: Adjoint and self-adjoint operators on Hilbert spaces (ver: 14.01.2021)
- PS11: Compact operators, spectral theory (ver: 28.01.2021)
- PS12: Convolutions, density of smooth functions and Schwartz spaces (ver: 28.01.2021)
- PS13: Fourier transform (ver: 28.01.2021)

** Homeworks ** are listed here

** Solutions to homeworks ** :

- Homework for 22.10 (Ola, Krzysiek, Adam, Janek Kostrzon)
- Homework for 29.10 (Mateusz, Kuba)
- Homework for 5.11 (Janek Kociniak, Wojtek)
- Homework for 12.11 (Asia, Janek Kostrzon)
- Homework for 19.11 (Janek Kociniak, Wojtek)
- Homework for 26.11 (Filip, Michał)
- Homework for 3.12 (Filip, Michał)
- Homework for 10.12 (Ola, Krzysiek)
- Homework for 17.12: Problem 1 (with comments) and Problem 2 (Janek, Wojtek)
- Homework for 7.01 (Janek Kociniak, Wojtek) and my comments on application of orthonormal basis
- Homework for 14.01 (Janek Kociniak, Wojtek)
- Homework for 21.01 (Ola, Krzysiek)
- Homework for 28.01 (Ola, Krzysiek)

** Zoom blackboards and YouTube links: ** :

- Class 1 (15.10): blackboard is here, YouTube link is here
- Class 2 (22.10): blackboard is here, YouTube link is here
- Class 3 (29.10): blackboard is here, YouTube link is here
- Class 4 (5.11): blackboard is here, YouTube link is here
- Class 5 (19.11): blackboard is here, YouTube link is here
- Class 6 (26.11): blackboard is here, YouTube link is here
- Class 7 (3.12): blackboard is here, YouTube link is here
- Class 8 (10.12): blackboard is here, YouTube link is here
- Class 9 (17.12): blackboard is here, YouTube link is here
- Class 10 (7.01): blackboard is here, YouTube link is here
- Class 11 (14.01): blackboard is here, YouTube link is here
- Class 12 (21.01): blackboard is here, YouTube link is here
- Class 13 (28.01): blackboard is here, YouTube link is here

** Group office hours: ** :

- 25.11.2020, 15:30-18:00: blackboard is here, YouTube link is here
- 12.01.2020, 15:30-18:00: blackboard is here, YouTube link is here
- 30.01.2020, 12:00-15:00: blackboard is here, YouTube link is here

** Class results (homework, activities) ** are updated here.

** Special Problems ** are here. Some solutions will be posted below.

- Problem 1 (on Sobolev spaces) by Patryk Szlufik,

** Others ** :

- Additional basic problems (PS1, PS2)
- Solutions to Problems 1, 4, 5 on the 1st midterm
- Dual spaces to spaces of bounded functions/sequences
- Solutions to Problems 2, 3 on the 2nd midterm

** Class journal ** :

- 28.01.2021: We discussed S1-S3 from PS12, 1, 2(A-C), 3-5 from PS13.
- 21.01.2021: We discussed B2, B4, C1 from PS11, A1-A3, B1, B3 from PS12.
- 14.01.2021: We discussed 4, 6, 13 from PS9; B5, C4 from PS10 and A1-A6, B2 from PS11.
- 7.01.2021: We discussed 1,2,3,5 from PS9; properties A1-A9 from PS10 and B1, B2, B4, C3 from PS10.
- 17.12.2020: We discussed 1,2,3,5,6,8,13 from PS8, B4/PS7 and A7/PS6. Merry Christmas!
- 10.12.2020: We discussed A4-A8 and C5, C7 from PS6; A4, A5, B2, B3 from PS7.
- 3.12.2020: We discussed A1-A3, B1-B4, B7, C1-C3 from PS6 and A1-A3, B1 from PS7. Problems B8/PS6 and A6/PS7 form homework for next week.
- 26.11.2020: We discussed B8 from PS3, A4, A5, B5 from PS4, A4-A5, C1-C3 from PS5. We made a small review before the midterm.
- 19.11.2020: We discussed E1 from PS1, B5, B7 from PS3, B1-B3 from PS4, A1-A3 from PS5 and B1, B3, B6 from PS5.
- 5.11.2020: We discussed B14 from PS2, A1-A3, B1-B4 from PS3, A1-A2 from PS4 and E1 from PS1.
- 29.10.2020: We discussed D2 from PS1; B4, B6-B9, C1-C3, D1-D4 from PS2.
- 22.10.2020: We discussed C5-C6 from PS1 and A1-A4, B1-B2 from PS2. Homeworks consists of C7 (PS1) and D1 (PS1) + B3 (PS2).
- 15.10.2020: We discussed A1-A5, B1-B3 and C1-C4 from PS1 where B3 and C4 form your homework for next week.
- 2.10.2020: I sent e-mail with organizational stuff, see here.