MMA120 Functional Analysis Autumn 20
Course PM
- This page contains the program of the course: lectures, exercise sessions, assignments. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM. '
Online classes (Link, Schedule etc)
We shall have online classes with Zoom, the link is given below. All of us (might) have had online classes and please follow the general recommendations/rules about technical details on attending lectures, audio/video communications, examinations.
(1) ZOOM-LINK:
Topic: Functional Analysis
Time: This is a recurring meeting Meet anytime
Join from PC, Mac, Linux, iOS or Android: https://chalmers.zoom.us/j/63090882852
Password: 759793
Or iPhone one-tap :
Sweden: +46844682488,,63090882852# or +46850163827,,63090882852#
Or Telephone:
If you have problems with +46 7 6692 0434 in Sweden please dial +46 8 4468 2488 instead.
Dial(for higher quality, dial a number based on your current location):
Sweden: +46 8 4468 2488 or +46 8 5016 3827 or +46 8 5050 0828 or +46 8 5050 0829 or +46 8 5052 0017 or +46 850 539 728
Meeting ID: 630 9088 2852
Password: 759793
International numbers available: https://chalmers.zoom.us/u/ccAd7jk5eM
Or an H.323/SIP room system:
H.323: 109.105.112.236 or 109.105.112.235
Meeting ID: 630 9088 2852
Password: 759793
SIP: 63090882852@109.105.112.236 or 63090882852@109.105.112.235
Password: 759793
(2) Schedule, Week 45-Week 51 (Nov. 1- Dec. 18):
Tuesday 10:00-12:00,
Thursday 13:15 - 15:00,
Friday 08:00-10:00.
The oral examination will be in next year, Week 2, January 2021 and we shall fix the schedule in Week 51, Dec. 17-18.
Program
The schedule of the course is in TimeEdit.
Course literature: M. Einsiedler and T. Ward, Functional Analysis, Spectral Theory and Applications. Springer GTM.
Chapt. 2, 2.1—2.2. 2.4. (Section 2.2 on space of continuous functions and Stone - Weierstrass theorem is covered by the course Real Analysis, MMA120, by Ulla Dinger).
Chapt. 3, 3.1-3.2.
Chapt. 4, 4.1-4.2.
Chapt. 5, 5,1.
Chapt. 7, 7.1-7.4.
Chapt. 8. 4.1 (Banach-Alaoglu theorem), 4.2 (Applications of B-A theorem, part of)
Brief summary:
The course has roughly two parts, the "abstract" Functional Analysis, and the "concrete" Functional Analysis. The "abstract" part is to develop general theory for Banach spaces and operators, and it is in
Chapt. 2.1-2.2, Chapt 3.1, Chapt 4.
The "concrete" part is on the specialization and application of the abstract theory to concrete spaces and it is in
Chapt 2.3, 3.1.3, Chapt 7.
The plan of the course is that we cover the "abstract Functional Analysi" first and then the "concret Functional Analysis".
List of Theorems whose statements and proofs are required for the exam.:
- Proposition 2.35 (non-compactness)
- Theorem 3.13 (for convex sets in Hilbert spaces)
- Corollary 3.19. (Riesz representation)
- Theorem 4.1 (Banach-Steinhauss)
- Theorem 4.12 (Baire category)
- Theorem 4.28 (Closed graph)
- Hahn-Banach
- Existence of Banach limit on l ^ \infty (small l infinity)-
- Banach-Alaoglu Theorem
- Dual space of L^1.
Lectures (with a summary of completed sections)
Tuesdays and Thursdays classes will be lectures, and Fridays will be exercise classes and discussions.
Supplementary text/solutions to exercises:
(1) "Elementary" proof of Prop. 2.6 without using compactness
(3) Weak (weak*)-closure of the sphere.
(Lecture Notes will be uploaded under sections)
| Week |
Content |
Sections [EW] | Summary |
|---|---|---|---|
| 45 |
Introduction/Motivation. Normed vector spaces. Quotient of normed vector spaces |
2.1.1-2.2.2 (Lecture Notes will be uploaded here.) |
Normed vector spaces and examples. Quotient spaces and quotient semi-norms. Example of quotient semi-norms. |
| 46 |
Non-compactness of unit ball in infinite-dimensional Banach spaces. Bounded operators and linear functionals. (We leave 2.3.3 to later parts; the abstract Stone-Weirstrass theorem is not part of the course - we might have time later to prove it.) |
2.2.3-2.4.1 (Section 2.3.1 on Arzela-Ascoli theorem is moved to Week 49, it will be treated with other related compactness results.) (with plan for next week.) |
Norm equivalence of all finite dim. normed spaces. Topology of the closed unit ball in normed spaces. Definition of continuity of linear operators. |
| 47 | Hilbert spaces and generally uniformly convex spaces (such as l^p spaces for p in [2, \infty) |
2.4.2, 3.1.1-3.1.2. |
Orthogonal projection, proof without using basis. Riesz Representation Theory. Uniform convexity. |
| 48 | Banach-Steihauss and Baire Theorems. |
4.1,1-4.2.3. |
BS and Baire Thms. Application to Convergence of Fourier Series. |
| 49 |
Open Mapping and Closed Graph Theorems (CG) Hahn-Banach theorem (H-B).
|
4.2. 2.3.1. 7.1.1. |
CG, H-B and applications: Separation, Banach limit, bi-dual. |
| 50 |
Compactness results: Banach-Alaoglu for the closed unit ball in dual space X^*. Arzela-Ascoli. Dual spaces of L^p. (l^p dual was computed in week 47). Hölder inequality |
7.1.1-7.2.2, 7.3-7.4 Exercise-Class-Fri
|
B-A Theorem. A-A Theorem. Proof of Holder inequality and l_p-norm is a norm. |
| 51 | Dual spaces. Applications. Equi-distributions. |
2.3.3, 8.2.1, 7.2.3
|
Duals space of concrete spaces, Riesz Theorem for C(X)^*. Application of Riesz and B-A Theorems to Equi-distributions. |
| W2 |
Examination (Week 2, Thursday and Friday) |
Recommended exercises
(There are so called Essential Exercises in the book - please try to work them out. Exercises marked with * might be a bit difficult. Some extra exercises closely related to the lectures will be handed out together with the assignments.)
2.3a), 2.7, 2.9, 2.16, 2.18, 2.25, 2.26, 2.36*, 2.55, 2.56, 2.58
3.4, 3.5, 3.9, 3.10, 3.14 (1)(2), 3. 15 (Do this for R^n with l^p-norm first). 3.20, 3.27, 3.28*, 3.37. 3.40
4.4, 4.5, 4.16, 4.18, 4.20, 4.23, 4.29. 7.2, 7.7, 7.11, 7.33 (a)-(d),
7.56, 7.58. 8.6, 8.8, 8.9, 8.12*, 8.16, 8.17, 8.18.
Demonstration Exercises:
Week 45: (a) l^p-norm is a norm. (b) Different elementary proof of equivalence of norms on R^n. (c) Examples of unbounded linear functionals. (See Assignment-1 with extra exerecises).
Week 46: Exercise 2.16. Extra-Ex 7, 8.
Week 47: Exercise 3.15*. Exercises in Assignment No. 1.
Week 48: Assignment 2. Exercise 4.4. 4.20.
Week 49: Ex 2.36* (This exercise is a bit difficult. Please prepare and try it in advance. Try to read the next section on Arrzela-Ascoli theorem ), 3.28 (on c_0 being non-complemented subspace of l^\infty.) .
Week 50: Ex. 7.2, 7.7. (These are some kind of "standard" exercises.) Exercise 7.11 (see also the extra exercise 9 in assighment 1). Assignments 3 and 4
Week 51: Ex 8.8 (This is also some standard ex. Try to work it out on your own). Ex. 8.16, 8.18. Assignment 4 and 5.
Assignments
There will be 5 homework assignments, each will be graded with points and will be added up along with the oral examination to the final grade. They are in Canvas/Assignments where you can also upload your solutions.
Student representatives
The following students have been appointed:
Petar Jovanovski, petarj@chalmers.se
Lorents Landgren, lorents@student.chalmers.se
Petar, Lorents and I welcome all your critiques/feedbacks.
Course summary:
| Date | Details | Due |
|---|---|---|