WS 19/20: ** 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 ** :
Tutorial carries a weight of **15%**. This will be based on homework assignments (**10%**) (written, graded on scale 0, 0.5, 1) and active class participation (**5%**).
Problems discussed on tutorials will be uploaded in advance and it is highly recommended to think them over before the class. There are also ** Special Problems ** (see below) that can be used to earn additional points.

** Grading assistant ** is Mr. Bohdan Petraszczuk (bp372955@...).

** Problem Sets ** :

- Set 1: normed spaces, Banach spaces, L^p spaces
- Set 2: operators and their norms
- Set 3: Baire Category Theorem and Uniform Boundedness Principle
- Set 4: Open Mapping Theorem and Closed Graph Theorem
- Set 5: Introduction to Hilbert Spaces
- Revision set before Midterm 1 and more Hilbert spaces
- Set 6: Dual spaces, Hahn-Banach separation theorems and weak convergence
- Set 7: Radon-Nikodym Theorem and spaces of measures
- Set 8: Spectrum and adjoints on Hilbert spaces
- Set 9: Spectral theory of compact and self-adjoint operators
- Set 10: Convolutions and Schwartz spaces
- Set 11: Fourier series, Fourier transform and tempered distributions

** Big Homeworks ** :

- Big HW 1 (announced 17/10, deadline: 31/10)
- Big HW 2 (announced 31/10, deadline: 14/11)
- Big HW 3 (announced 14/11, deadline: 5/12)
- Big HW 4 (announced 5/12, deadline: 19/12)
- Christmas Problems (announced 20/12, deadline: 9/01)
- Big HW 5 (announced 9/01, deadline: 23/01)

** Solutions to homeworks ** :

- Homework for 10.10.2019
- Homework for 17.10.2019 and my comments
- Homework for 24.10.2019
- Homework for 30.10.2019
- Big Homework 1 for 30.10.2019
- Big Homework 2 for 14.11.2019
- Homework for 21.11.2019 with Hahn-Banach and without
- Homework for 2.12.2019
- Big Homework 3 for 5.12.2019: Your solutions to 1st, 2nd, 5th and my to 3rd and 4th problem
- Homework for 12.12.2019
- Big Homework 4 for 19.12.2019: Your solutions to 1st, 3rd and my to 2nd and 4th problem
- Christmas Problems solutions and my comments
- Big Homework 5 for 23.01.2020

** Notes ** :

- Problem Set 1 (hand-written notes, 17.10.2019)
- Problem Set 2 (hand-written notes, 24.10.2019)
- Problem Set 3 (hand-written notes, 24.10.2019)
- Problem Set 4 (hand-written notes, 24.10.2019)
- Problem Set 5 (hand-written notes, 07.11.2019)
- Review Set + some Hilbert spaces (hand-written notes, 07.11.2019)
- Problem Set 6 (hand-written notes, 2.12.2019)
- Problem Set 7 (hand-written notes, 2.12.2019)
- Problem Set 8 (hand-written notes, 12.12.2019)
- Problem Set 9 (hand-written notes, 19.12.2019)
- Summary on techniques used to study spectrum
- Problem Set 10 (hand-written notes, 9.01.2020)
- Problem Set 11 (hand-written notes, 27.01.2020)

** Homeworks results ** are
here.

** Special Problems ** are
here. Some solutions:

** Class journal ** :

- ∞ ∞ ∞ ∞: The winner of Special Problems Competition is
**Szymon ZWARA**. Congratulations to him and all the people solving special problems! - 28.01.2020: As arranged with some of you, office hours before exam are on Sunday, 2nd Feb at 4pm in room 4040.
- 27.01.2020: We discussed T3-T16 from PS11. Next week there is final exam. Good luck!
- 16.01.2020: We discussed 2nd problem from the midterm concerning shift operator on L^2 (see solution ). Then, we discussed S1, S4, S5, S6, S7 and P1 from PS10. Finally, we moved to Fourier series and introduced their history and some convergence results (S1-S8 from PS11). We started to study Fourier transform (T1, T2 from PS11).
**IMPORTANT:**There is no class on Thr 23/01. Additional tutorial is scheduled for Mon 27/01, 10am in the room 3140.**HOMEWORK SUBMISSION:**Put them in the envelope in front of my office 4040 at most on Saturday 25/01 around 12am. - 9.01.2020: Plan for the class is here . We have discussed Problem 4 from BH3, Problem 2 from BH4, Christmas Problems. We have reviewed methods used to study spectrum as well as spectral theory for compact and for self-adjoint operators. Then, we studied spectral decomposition for compact and self-adjoint operators (ST2 from PS9). Then, we discussed convolutions and Schwarz spaces: C1, C2, C4, C5, C6, S2, S3 from PS10. The last Big Homework 5 has been announced with two deadlines: 16.01 for Problem 1 and 2, 23.01 for Problem 3 and 4. Tomorrow is the 2nd midterm. Good luck!
- 6.01.2020: As arranged with some of you, additional office hours before 2nd midterm will be on
**Wed 8/01 (4 pm, room 5070)**and**Thr 9/01 (around 5 pm, room to be confirmed)**. - 19.12.2019: Note carefully errors on PS9!! We discussed S12, S15, A1, A2 from PS8 and S1-S5, C1-C10 from PS9. There is "Christmas HW" with deadline 9/01 but if you want it graded before 9/01, you can leave it in the envelope (room 4040) at most on 8/01. There are two new special problems. Merry Christmas!
- 12.12.2019: We discussed S6, S7, S8, S9, S10(a), S11 and S15 from PS8. There is no small homework this week but there is Big Homework 4 due to 19/12/2019. Note that all problems in BH4 are relevant to the second midterm. I added a small hint to Problem 4 in BH4.
- 5.12.2019: We started PS8 today and we discussed R1-R8, M1-M7 and S1-S5. Small homework (deadline: 12/12) is R2. Big Homework 4 was announced today.
- 2.12.2019: We had additional tutorial today. We discussed W9 and W10 from PS6 - in particular we proved Banach-Alaoglu-Bourbaki for separable Hilbert spaces. Then, we discussed PS7. This includes Radon - Nikodym Theorem and topological properties of space of bounded measures (normed Banach space but not separable in general). We observed some continuity pathology for total variation distance. Therefore, we introduced Wasserstein metric that is well-known in optimal transport theory.
**Small homework for today:**some of you did not submitted it. This problem is very important and I strongly encourage you to solve it anyway.**Comment on the content of Problem Sets:**I constantly update Problem Sets (there are some errors...) and in particular the printed version provided in the class may contain some errors / may lack some problems. - 21.11.2019:
**IMPORTANT: There is no class on Thursday 28/11. We meet for an additional tutorial on Monday 2/12, room 3140 (the same room as the regular class), 10:15-11:45.** - 21.11.2019: We finished Hahn-Banach theorems (H11, H12, H14, H16), weak convergence (W1-W7) and we started talking about spaces of measures and Radon-Nikodym Theorem (R1 from the new PS7). Small homework for application of Hahn-Banach is W7 (deadline: 2/12). Note that the deadline for Big Homework 3 is 5/12.
- 20.11.2019: There are some new Special Problems (4,5,6) concerning weak convergence. I strongly encourage those of you who think seriously about further studies in Analysis, PDEs or Probability to consider these problems obligatory.
- 14.11.2019: We discussed duality representations (D1-D3), Hahn-Banach theorems (H1-H10, H13 and H15) and we started discussion about weak topologies (W4) and motivation for introducing them (see notes). Small homework is Riesz Lemma H13 (deadline: 21/11). There is new Big Homework.
- 9.11.2019: New Special Problems have been announced. As arranged before, there are office hours before midterm: Saturday 9/11 (romm 4040, 16-19) and Wednesday 13/11 (room 5070, 16-19).
- 7.11.2019: We finished discussion about complement of c0 in l^∞ (C3 from PS5). Then, we discussed C5 (from PS4, that was homework with extended deadline). Finally, we finished PS5 (R1-R4 and B1-B4). Then, we discussed H1-H6 from midterm set. There are also some problems (R1-R4) that you can solve by yourself as some midterm practise (answers are provided in the notes). Note that next week there is a deadline for Big Homework 2 which is also relevant for midterm exam.
**There is no small homework for this week.** - 31.10.2019: We discussed I4 (PS2 or PS4), P1-P8, O1-09 (PS5). We have started discussing why c0 has no complement in l^∞ (C1 and C2 from PS5) - review this carefully as we conclude the proof next time. Small homework is P8 (from PS5; deadline 7/11). Deadline for small homework for today was extended until 7/11 (there is also a hint: "consider sequence with a very small parameter").
**Congratulations to Szymek Antoniak who solved the problem without a hint!!!**Big Homework 2 was announced today with deadline 14/11. - 24.10.2019: We discussed U2, U3, U6 (PS3), the whole PS4 and I1, I2 from PS2 (note that I1-I4 are copied in PS4 for convenience). Small homework is C5 (from PS4; deadline 31/10). Note that there is a deadline for Big Homework 1 next week (31/10).
- 19.10.2019: I uploaded solutions to 2nd homework (author: Kuba Woźnicki) and some comments - it is strongly recommended to read them carefully. Problem 3 in Big Homework has been formulated more precisely (due to Szymon's question).
- 17.10.2019: We discussed C2, C4, C5 (PS1), A1, A2, A4-A6, N7, N8 (PS2), B1, B2, B5, U4, U5 (PS3). Small homework is U5 from PS3 (deadline: 24/10/2019). The first big homework was announced today (see above) with deadline 31/10. It contains problems C5 (PS1) and A4, N8 (PS2). Note that Problem Set 2 has been updated today (see Problem A2).
- 10.10.2019: We discussed S3-S7, A1 from PS1 and F1-F4, N1-N3 from PS2. Homework (written): S4, S7 (from PS1) and N2 (from PS2).
- 3.10.2019: I am out of Poland for the period 5.10 - 12.10. Nevertheless, tutorial will be offered on the regular basis.
- 3.10.2019: We discussed L1-L6, L8, C3, S1-S2 from PS1. Homework (written): L1, L6, C3.